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(Not an answer, but a long addendum to the question above to prevent clutter.)

Here are the $2+23+1 = 26\,$ "conjugates": as $2$ singletons, $23$ pairs, and $1$ quadruplet. As requested in the comments, the Wythoff symbol is included. The table is divided into symbols of $$\text{Form}\,1:\; p|q\;r\\ \text{Form}\,2:\; p\;q|r\\ \text{Form}\,3:\; p\;q\;r|\\ \text{Form}\,4:\; |p\;q\;r$$ It turns out that if a uniform polyhedron has Wythoff symbol of $\text{Form}\,n$, then its "conjugate" also has symbol of $\text{Form}\,n$. A convenient list of $U_n$ is here.

$$\begin{array}{|c|c|c|} \hline \text{Number}&\text{Polyhedron}&\text{Wythoff symbol}\\ \hline 1 & U_{22}\;\text{(icosahedron)} & 5|2\;3\\ 1 & U_{53} & 5/2|2\;3\\ 2 & U_{23}\;\text{(dodecahedron)} & 3|2\;5\\ 2 & U_{52} & 3|2\;5/2\\ 3 & U_{24} & 2|3\;5\\ 3 & U_{54} & 2|5/2\;3\\ 4 & U_{34} & 5|2\; 5/2\\ 4 & U_{35} & 5/2|2\; 5\\ \hline 5 & U_{10} & 3\;4|2\\ 5 & U_{17} & 3/2\;4|2\\ 6 & U_{27} & 3\;5|2\\ 6 & U_{67} & 5/3\;3|2\\ 7 & U_{25} & 2\;5|3\\ 7 & U_{55} & 2\;5/2|3\\ 8 & U_{31} & 5/2\;3|3\\ 8 & U_{48} & 3/2\;5|3\\ 9 & U_{9} & 2\;3|4\\ 9 & U_{19} & 2\;3|4/3\\ 10 & U_{13} & 3/2\;4|4\\ 10 & U_{14} & 3\;4|4/3\\ 11 & U_{26} & 2\;3|5\\ 11 & U_{66} & 2\;3|5/3\\ 12 & U_{37} & 2\;5/2|5\\ 12 & U_{58} & 2\;5|5/3\\ 13 & U_{42} & 3\;5|5/3\\ 13 & U_{43} & 5/3\;3|5\\ 14 & U_{33} & 3/2\;5|5\\ 14 & U_{61} & 5/2\;3;5/3\\ 15 & U_{49} & 3/2\;3|5\\ 15 & U_{71} & 3/2\;3|5/3\\ 16 & U_{51} & 5/4\;5|5\\ 16 & U_{70} & 5/3\;5/2|5/3\\ \hline 17 & U_{11} & 2\;3\;4|\\ 17 & U_{20} & 4/3\;2\;3|\\ 18 & U_{28} & 2\;3\;5|\\ 18 & U_{68} & 5/3\;2\;3|\\ 19 & U_{18} & 3/2\;2\;4|\\ 19 & U_{21} & 4/3\;3/2\;2|\\ 20 & U_{50} & 3/2\;3\;5|\\ 20 & U_{63} & 5/3\;5/2\;3|\\ 21 & U_{39} & 2\;5/2\;5|\\ 21 & U_{73} & 3/2\;5/3\;2|\\ \hline \color{blue}{22} & U_{12} & |2\;3\;4\\ \color{brown}{23} & U_{46} & |5/3\;3\;5\\ 24 & U_{32} & |5/2\;3\;3\\ 24 & U_{72} & |3/2\;3/2\;5/2\\ 25 & U_{40} & |2\;5/2\;5\\ 25 & U_{60} & |5/3\;2\;5\\ \color{green}{26} & U_{29} & |2\;3\;5\\ \color{green}{26} & U_{57} & |2\;5/2\;3\\ \color{green}{26} & U_{69} & |5/3\;2\;3\\ \color{green}{26} & U_{74} & |3/2\;5/3\;2\\ \hline \end{array}$$

Notes :

1. All these $52$ $U_n$ have $R_n^2$ that is an algebraic number of degree $k>1$. But the $U_n$ with $|p\;q\;r$ have degree $k\geq3$.
2. The special case of $R_{12}^2$ and $R_{46}^2$ (for the snub cube and snub icosidodecadodecahedron) use cubics with one real root, namely the well-known tribonacci constant and plastic constant.
3. Two pairs of $R_n^2$ are quartic roots.
4. Four $R_n^2$ use the same sextic with four real roots.

(Not an answer, but a long addendum to the question above to prevent clutter.)

Here are the $2+23+1 = 26\,$ "conjugates": as $2$ singletons, $23$ pairs, and $1$ quadruplet. As requested in the comments, the Wythoff symbol is included. The table is divided into symbols of $$\text{Form}\,1:\; p|q\;r\\ \text{Form}\,2:\; p\;q|r\\ \text{Form}\,3:\; p\;q\;r|\\ \text{Form}\,4:\; |p\;q\;r$$ It turns out that if a uniform polyhedron has Wythoff symbol of $\text{Form}\,n$, then its "conjugate" also has symbol of $\text{Form}\,n$. A convenient list of $U_n$ is here.

\begin{array}{|c|c|rl|} \hline \text{Number}&\text{Polyhedron}&\text{Wythoff}&\!\!\!\text{symbol}\\ \hline 1 & U_{22}\;\text{(icosahedron)} & 5&|\ 2\;3\\ 1 & U_{53} & 5/2&|\ 2\;3\\ 2 & U_{23}\;\text{(dodecahedron)} & 3&|\ 2\;5\\ 2 & U_{52} & 3&|\ 2\;5/2\\ 3 & U_{24} & 2&|\ 3\;5\\ 3 & U_{54} & 2&|\ 5/2\;3\\ 4 & U_{34} & 5&|\ 2\; 5/2\\ 4 & U_{35} & 5/2&|\ 2\; 5\\ \hline 5 & U_{10} & 3\;4&|\ 2\\ 5 & U_{17} & 3/2\;4&|\ 2\\ 6 & U_{27} & 3\;5&|\ 2\\ 6 & U_{67} & 5/3\;3&|\ 2\\ 7 & U_{25} & 2\;5&|\ 3\\ 7 & U_{55} & 2\;5/2&|\ 3\\ 8 & U_{31} & 5/2\;3&|\ 3\\ 8 & U_{48} & 3/2\;5&|\ 3\\ 9 & U_{9} & 2\;3&|\ 4\\ 9 & U_{19} & 2\;3&|\ 4/3\\ 10 & U_{13} & 3/2\;4&|\ 4\\ 10 & U_{14} & 3\;4&|\ 4/3\\ 11 & U_{26} & 2\;3&|\ 5\\ 11 & U_{66} & 2\;3&|\ 5/3\\ 12 & U_{37} & 2\;5/2&|\ 5\\ 12 & U_{58} & 2\;5&|\ 5/3\\ 13 & U_{42} & 3\;5&|\ 5/3\\ 13 & U_{43} & 5/3\;3&|\ 5\\ 14 & U_{33} & 3/2\;5&|\ 5\\ 14 & U_{61} & 5/2\;3&|\ 5/3\\ 15 & U_{49} & 3/2\;3&|\ 5\\ 15 & U_{71} & 3/2\;3&|\ 5/3\\ 16 & U_{51} & 5/4\;5&|\ 5\\ 16 & U_{70} & 5/3\;5/2&|\ 5/3\\ \hline 17 & U_{11} & 2\;3\;4&| \\ 17 & U_{20} & 4/3\;2\;3&| \\ 18 & U_{28} & 2\;3\;5&| \\ 18 & U_{68} & 5/3\;2\;3&| \\ 19 & U_{18} & 3/2\;2\;4&| \\ 19 & U_{21} & 4/3\;3/2\;2&| \\ 20 & U_{50} & 3/2\;3\;5&| \\ 20 & U_{63} & 5/3\;5/2\;3&| \\ 21 & U_{39} & 2\;5/2\;5&| \\ 21 & U_{73} & 3/2\;5/3\;2&| \\ \hline \color{blue}{22} & U_{12} & &|\ 2\;3\;4\\ \color{brown}{23} & U_{46} & &|\ 5/3\;3\;5\\ 24 & U_{32} & &|\ 5/2\;3\;3\\ 24 & U_{72} & &|\ 3/2\;3/2\;5/2\\ 25 & U_{40} & &|\ 2\;5/2\;5\\ 25 & U_{60} & &|\ 5/3\;2\;5\\ \color{green}{26} & U_{29} & &|\ 2\;3\;5\\ \color{green}{26} & U_{57} & &|\ 2\;5/2\;3\\ \color{green}{26} & U_{69} & &|\ 5/3\;2\;3\\ \color{green}{26} & U_{74} & &|\ 3/2\;5/3\;2 \\ \hline \end{array}

Notes :

1. All these $52$ $U_n$ have $R_n^2$ that is an algebraic number of degree $k>1$. But the $U_n$ with $|p\;q\;r$ have degree $k\geq3$.
2. The special case of $R_{12}^2$ and $R_{46}^2$ (for the snub cube and snub icosidodecadodecahedron) use cubics with one real root, namely the well-known tribonacci constant and plastic constant.
3. Two pairs of $R_n^2$ are quartic roots.
4. Four $R_n^2$ use the same sextic with four real roots.

(Not an answer, but a long addendum to the question above to prevent clutter.)

Here are the $2+23+1 = 26\,$ "conjugates": as $2$ singletons, $23$ pairs, and $1$ quadruplet. As requested in the comments, the Wythoff symbol is included. The table is divided into symbols of $$\text{Form}\,1:\; p|q\;r\\ \text{Form}\,2:\; p\;q|r\\ \text{Form}\,3:\; p\;q\;r|\\ \text{Form}\,4:\; |p\;q\;r$$ It turns out that if a uniform polyhedron has Wythoff symbol of $\text{Form}\,n$, then its "conjugate" also has symbol of $\text{Form}\,n$. A convenient list of $U_n$ is here.

$$\begin{array}{|c|c|c|} \hline \text{Number}&\text{Polyhedron}&\text{Wythoff symbol}\\ \hline 1 & U_{22}\;\text{(icosahedron)} & 5|2\;3\\ 1 & U_{53} & 5/2|2\;3\\ 2 & U_{23}\;\text{(dodecahedron)} & 3|2\;5\\ 2 & U_{52} & 3|2\;5/2\\ 3 & U_{24} & 2|3\;5\\ 3 & U_{54} & 2|5/2\;3\\ 4 & U_{34} & 5|2\; 5/2\\ 4 & U_{35} & 5/2|2\; 5\\ \hline 5 & U_{10} & 3\;4|2\\ 5 & U_{17} & 3/2\;4|2\\ 6 & U_{27} & 3\;5|2\\ 6 & U_{67} & 5/3\;3|2\\ 7 & U_{25} & 2\;5|3\\ 7 & U_{55} & 2\;5/2|3\\ 8 & U_{31} & 5/2\;3|3\\ 8 & U_{48} & 3/2\;5|3\\ 9 & U_{9} & 2\;3|4\\ 9 & U_{19} & 2\;3|4/3\\ 10 & U_{13} & 3/2\;4|4\\ 10 & U_{14} & 3\;4|4/3\\ 11 & U_{26} & 2\;3|5\\ 11 & U_{66} & 2\;3|5/3\\ 12 & U_{37} & 2\;5/2|5\\ 12 & U_{58} & 2\;5|5/3\\ 13 & U_{42} & 3\;5|5/3\\ 13 & U_{43} & 5/3\;3|5\\ 14 & U_{33} & 3/2\;5|5\\ 14 & U_{61} & 5/2\;3;5/3\\ 15 & U_{49} & 3/2\;3|5\\ 15 & U_{71} & 3/2\;3|5/3\\ 16 & U_{51} & 5/4\;5|5\\ 16 & U_{70} & 5/3\;5/2|5/3\\ \hline 17 & U_{11} & 2\;3\;4|\\ 17 & U_{20} & 4/3\;2\;3|\\ 18 & U_{28} & 2\;3\;5|\\ 18 & U_{68} & 5/3\;2\;3|\\ 19 & U_{18} & 3/2\;2\;4|\\ 19 & U_{21} & 4/3\;3/2\;2|\\ 20 & U_{50} & 3/2\;3\;5|\\ 20 & U_{63} & 5/3\;5/2\;3|\\ 21 & U_{39} & 2\;5/2\;5|\\ 21 & U_{73} & 3/2\;5/3\;2|\\ \hline \color{blue}{22} & U_{12} & |2\;3\;4\\ \color{brown}{23} & U_{46} & |5/3\;3\;5\\ 24 & U_{32} & |5/2\;3\;3\\ 24 & U_{72} & |3/2\;3/2\;5/2\\ 25 & U_{40} & |2\;5/2\;5\\ 25 & U_{60} & |5/3\;2\;5\\ \color{green}{26} & U_{29} & |2\;3\;5\\ \color{green}{26} & U_{57} & |2\;5/2\;3\\ \color{green}{26} & U_{69} & |5/3\;2\;3\\ \color{green}{26} & U_{74} & |3/2\;5/3\;2\\ \hline \end{array}$$

Notes :

1. The $U_n$ with form $|p\;q\;r$ have $R_n^2$ that is an algebraic number of degree $k\geq3$.
2. The special case of $R_{12}^2$ and $R_{46}^2$ (for the snub cube and snub icosidodecadodecahedron) use cubics with one real root, namely the well-known tribonacci constant and plastic constant.
3. Two pairs of $R_n^2$ are quartic roots.
4. Four $R_n^2$ use the same sextic with four real roots.

(Not an answer, but a long addendum to the question above to prevent clutter.)

Here are the $2+23+1 = 26\,$ "conjugates": as $2$ singletons, $23$ pairs, and $1$ quadruplet. As requested in the comments, the Wythoff symbol is included. The table is divided into symbols of $$\text{Form}\,1:\; p|q\;r\\ \text{Form}\,2:\; p\;q|r\\ \text{Form}\,3:\; p\;q\;r|\\ \text{Form}\,4:\; |p\;q\;r$$ It turns out that if a uniform polyhedron has Wythoff symbol of $\text{Form}\,n$, then its "conjugate" also has symbol of $\text{Form}\,n$. A convenient list of $U_n$ is here.

$$\begin{array}{|c|c|c|} \hline \text{Number}&\text{Polyhedron}&\text{Wythoff symbol}\\ \hline 1 & U_{22}\;\text{(icosahedron)} & 5|2\;3\\ 1 & U_{53} & 5/2|2\;3\\ 2 & U_{23}\;\text{(dodecahedron)} & 3|2\;5\\ 2 & U_{52} & 3|2\;5/2\\ 3 & U_{24} & 2|3\;5\\ 3 & U_{54} & 2|5/2\;3\\ 4 & U_{34} & 5|2\; 5/2\\ 4 & U_{35} & 5/2|2\; 5\\ \hline 5 & U_{10} & 3\;4|2\\ 5 & U_{17} & 3/2\;4|2\\ 6 & U_{27} & 3\;5|2\\ 6 & U_{67} & 5/3\;3|2\\ 7 & U_{25} & 2\;5|3\\ 7 & U_{55} & 2\;5/2|3\\ 8 & U_{31} & 5/2\;3|3\\ 8 & U_{48} & 3/2\;5|3\\ 9 & U_{9} & 2\;3|4\\ 9 & U_{19} & 2\;3|4/3\\ 10 & U_{13} & 3/2\;4|4\\ 10 & U_{14} & 3\;4|4/3\\ 11 & U_{26} & 2\;3|5\\ 11 & U_{66} & 2\;3|5/3\\ 12 & U_{37} & 2\;5/2|5\\ 12 & U_{58} & 2\;5|5/3\\ 13 & U_{42} & 3\;5|5/3\\ 13 & U_{43} & 5/3\;3|5\\ 14 & U_{33} & 3/2\;5|5\\ 14 & U_{61} & 5/2\;3;5/3\\ 15 & U_{49} & 3/2\;3|5\\ 15 & U_{71} & 3/2\;3|5/3\\ 16 & U_{51} & 5/4\;5|5\\ 16 & U_{70} & 5/3\;5/2|5/3\\ \hline 17 & U_{11} & 2\;3\;4|\\ 17 & U_{20} & 4/3\;2\;3|\\ 18 & U_{28} & 2\;3\;5|\\ 18 & U_{68} & 5/3\;2\;3|\\ 19 & U_{18} & 3/2\;2\;4|\\ 19 & U_{21} & 4/3\;3/2\;2|\\ 20 & U_{50} & 3/2\;3\;5|\\ 20 & U_{63} & 5/3\;5/2\;3|\\ 21 & U_{39} & 2\;5/2\;5|\\ 21 & U_{73} & 3/2\;5/3\;2|\\ \hline \color{blue}{22} & U_{12} & |2\;3\;4\\ \color{brown}{23} & U_{46} & |5/3\;3\;5\\ 24 & U_{32} & |5/2\;3\;3\\ 24 & U_{72} & |3/2\;3/2\;5/2\\ 25 & U_{40} & |2\;5/2\;5\\ 25 & U_{60} & |5/3\;2\;5\\ \color{green}{26} & U_{29} & |2\;3\;5\\ \color{green}{26} & U_{57} & |2\;5/2\;3\\ \color{green}{26} & U_{69} & |5/3\;2\;3\\ \color{green}{26} & U_{74} & |3/2\;5/3\;2\\ \hline \end{array}$$

Notes :

1. All these $52$ $U_n$ have $R_n^2$ that is an algebraic number of degree $k>1$. But the $U_n$ with $|p\;q\;r$ have degree $k\geq3$.
2. The special case of $R_{12}^2$ and $R_{46}^2$ (for the snub cube and snub icosidodecadodecahedron) use cubics with one real root, namely the well-known tribonacci constant and plastic constant.
3. Two pairs of $R_n^2$ are quartic roots.
4. Four $R_n^2$ use the same sextic with four real roots.

(Not an answer, but a long addendum to the question above to prevent clutter.)

Here are the $2+23+1 = 26\,$ "conjugates": as $2$ singletons, $23$ pairs, and $1$ quadruplet. As requested in the comments, the Wythoff symbol is included. The table is divided into symbols of $$\text{Form}\,1:\; p|q\;r\\ \text{Form}\,2:\; p\;q|r\\ \text{Form}\,3:\; p\;q\;r|\\ \text{Form}\,4:\; |p\;q\;r$$ It turns out that if a uniform polyhedron has Wythoff symbol of $\text{Form}\,n$, then its "conjugate" also has symbol of $\text{Form}\,n$.

$$\begin{array}{|c|c|c|} \hline \text{Number}&\text{Polyhedron}&\text{Wythoff symbol}\\ \hline 1 & U_{22}\;\text{(icosahedron)} & 5|2\;3\\ 1 & U_{53} & 5/2|2\;3\\ 2 & U_{23}\;\text{(dodecahedron)} & 3|2\;5\\ 2 & U_{52} & 3|2\;5/2\\ 3 & U_{24} & 2|3\;5\\ 3 & U_{54} & 2|5/2\;3\\ 4 & U_{34} & 5|2\; 5/2\\ 4 & U_{35} & 5/2|2\; 5\\ \hline 5 & U_{10} & 3\;4|2\\ 5 & U_{17} & 3/2\;4|2\\ 6 & U_{27} & 3\;5|2\\ 6 & U_{67} & 5/3\;3|2\\ 7 & U_{25} & 2\;5|3\\ 7 & U_{55} & 2\;5/2|3\\ 8 & U_{31} & 5/2\;3|3\\ 8 & U_{48} & 3/2\;5|3\\ 9 & U_{9} & 2\;3|4\\ 9 & U_{19} & 2\;3|4/3\\ 10 & U_{13} & 3/2\;4|4\\ 10 & U_{14} & 3\;4|4/3\\ 11 & U_{26} & 2\;3|5\\ 11 & U_{66} & 2\;3|5/3\\ 12 & U_{37} & 2\;5/2|5\\ 12 & U_{58} & 2\;5|5/3\\ 13 & U_{42} & 3\;5|5/3\\ 13 & U_{43} & 5/3\;3|5\\ 14 & U_{33} & 3/2\;5|5\\ 14 & U_{61} & 5/2\;3;5/3\\ 15 & U_{49} & 3/2\;3|5\\ 15 & U_{71} & 3/2\;3|5/3\\ 16 & U_{51} & 5/4\;5|5\\ 16 & U_{70} & 5/3\;5/2|5/3\\ \hline 17 & U_{11} & 2\;3\;4|\\ 17 & U_{20} & 4/3\;2\;3|\\ 18 & U_{28} & 2\;3\;5|\\ 18 & U_{68} & 5/3\;2\;3|\\ 19 & U_{18} & 3/2\;2\;4|\\ 19 & U_{21} & 4/3\;3/2\;2|\\ 20 & U_{50} & 3/2\;3\;5|\\ 20 & U_{63} & 5/3\;5/2\;3|\\ 21 & U_{39} & 2\;5/2\;5|\\ 21 & U_{73} & 3/2\;5/3\;2|\\ \hline \color{blue}{22} & U_{12} & |2\;3\;4\\ \color{brown}{23} & U_{46} & |5/3\;3\;5\\ 24 & U_{32} & |5/2\;3\;3\\ 24 & U_{72} & |3/2\;3/2\;5/2\\ 25 & U_{40} & |2\;5/2\;5\\ 25 & U_{60} & |5/3\;2\;5\\ \color{green}{26} & U_{29} & |2\;3\;5\\ \color{green}{26} & U_{57} & |2\;5/2\;3\\ \color{green}{26} & U_{69} & |5/3\;2\;3\\ \color{green}{26} & U_{74} & |3/2\;5/3\;2\\ \hline \end{array}$$

Notes :

1. The $U_n$ with form $|p\;q\;r$ have $R_n^2$ that is an algebraic number of degree $k\geq3$.
2. The special case of $R_{12}^2$ and $R_{46}^2$ (for the snub cube and snub icosidodecadodecahedron) use cubics with one real root, namely the well-known tribonacci constant and plastic constant.
3. Two pairs of $R_n^2$ are quartic roots.
4. Four $R_n^2$ use the same sextic with four real roots.

(Not an answer, but a long addendum to the question above to prevent clutter.)

Here are the $2+23+1 = 26\,$ "conjugates": as $2$ singletons, $23$ pairs, and $1$ quadruplet. As requested in the comments, the Wythoff symbol is included. The table is divided into symbols of $$\text{Form}\,1:\; p|q\;r\\ \text{Form}\,2:\; p\;q|r\\ \text{Form}\,3:\; p\;q\;r|\\ \text{Form}\,4:\; |p\;q\;r$$ It turns out that if a uniform polyhedron has Wythoff symbol of $\text{Form}\,n$, then its "conjugate" also has symbol of $\text{Form}\,n$. A convenient list of $U_n$ is here.

$$\begin{array}{|c|c|c|} \hline \text{Number}&\text{Polyhedron}&\text{Wythoff symbol}\\ \hline 1 & U_{22}\;\text{(icosahedron)} & 5|2\;3\\ 1 & U_{53} & 5/2|2\;3\\ 2 & U_{23}\;\text{(dodecahedron)} & 3|2\;5\\ 2 & U_{52} & 3|2\;5/2\\ 3 & U_{24} & 2|3\;5\\ 3 & U_{54} & 2|5/2\;3\\ 4 & U_{34} & 5|2\; 5/2\\ 4 & U_{35} & 5/2|2\; 5\\ \hline 5 & U_{10} & 3\;4|2\\ 5 & U_{17} & 3/2\;4|2\\ 6 & U_{27} & 3\;5|2\\ 6 & U_{67} & 5/3\;3|2\\ 7 & U_{25} & 2\;5|3\\ 7 & U_{55} & 2\;5/2|3\\ 8 & U_{31} & 5/2\;3|3\\ 8 & U_{48} & 3/2\;5|3\\ 9 & U_{9} & 2\;3|4\\ 9 & U_{19} & 2\;3|4/3\\ 10 & U_{13} & 3/2\;4|4\\ 10 & U_{14} & 3\;4|4/3\\ 11 & U_{26} & 2\;3|5\\ 11 & U_{66} & 2\;3|5/3\\ 12 & U_{37} & 2\;5/2|5\\ 12 & U_{58} & 2\;5|5/3\\ 13 & U_{42} & 3\;5|5/3\\ 13 & U_{43} & 5/3\;3|5\\ 14 & U_{33} & 3/2\;5|5\\ 14 & U_{61} & 5/2\;3;5/3\\ 15 & U_{49} & 3/2\;3|5\\ 15 & U_{71} & 3/2\;3|5/3\\ 16 & U_{51} & 5/4\;5|5\\ 16 & U_{70} & 5/3\;5/2|5/3\\ \hline 17 & U_{11} & 2\;3\;4|\\ 17 & U_{20} & 4/3\;2\;3|\\ 18 & U_{28} & 2\;3\;5|\\ 18 & U_{68} & 5/3\;2\;3|\\ 19 & U_{18} & 3/2\;2\;4|\\ 19 & U_{21} & 4/3\;3/2\;2|\\ 20 & U_{50} & 3/2\;3\;5|\\ 20 & U_{63} & 5/3\;5/2\;3|\\ 21 & U_{39} & 2\;5/2\;5|\\ 21 & U_{73} & 3/2\;5/3\;2|\\ \hline \color{blue}{22} & U_{12} & |2\;3\;4\\ \color{brown}{23} & U_{46} & |5/3\;3\;5\\ 24 & U_{32} & |5/2\;3\;3\\ 24 & U_{72} & |3/2\;3/2\;5/2\\ 25 & U_{40} & |2\;5/2\;5\\ 25 & U_{60} & |5/3\;2\;5\\ \color{green}{26} & U_{29} & |2\;3\;5\\ \color{green}{26} & U_{57} & |2\;5/2\;3\\ \color{green}{26} & U_{69} & |5/3\;2\;3\\ \color{green}{26} & U_{74} & |3/2\;5/3\;2\\ \hline \end{array}$$

Notes :

1. The $U_n$ with form $|p\;q\;r$ have $R_n^2$ that is an algebraic number of degree $k\geq3$.
2. The special case of $R_{12}^2$ and $R_{46}^2$ (for the snub cube and snub icosidodecadodecahedron) use cubics with one real root, namely the well-known tribonacci constant and plastic constant.
3. Two pairs of $R_n^2$ are quartic roots.
4. Four $R_n^2$ use the same sextic with four real roots.