This question--pertaining to the quantum-information-theoretic topic of "bound entanglement"--stems from the question and answer to https://math.stackexchange.com/questions/3464105/if-possible-meaningfully-simplify-an-expression-involving-logs-polylogs-and-hy/3465129#3465129

The answer (that is, the formula for the "bound entanglement probability") there contains the expression \begin{equation} \text{Li}_2\left(\frac{1}{18} \left(9+\sqrt{81-\frac{64}{\sqrt{3}}}\right)\right)-\text{Li}_2\left(\frac{1}{18} \left(9-\sqrt{81-\frac{64}{\sqrt{3}}}\right)\right), \end{equation} where the polylogarithm (in particular, dilogarithm) is indicated.

We have further observed as part of the simplification analysis in that answer--changing the subscript of Li from 2 to 1 (leading to the standard logarithmic framework)--that \begin{equation} \text{Li}_1\left(\frac{1}{18} \left(9+\sqrt{81-\frac{64}{\sqrt{3}}}\right)\right)-\text{Li}_1\left(\frac{1}{18} \left(9-\sqrt{81-\frac{64}{\sqrt{3}}}\right)\right) = \end{equation} \begin{equation} \log \left(\frac{1}{18} \left(9+\sqrt{81-\frac{64}{\sqrt{3}}}\right)\right)-\log \left(\frac{1}{18} \left(9-\sqrt{81-\frac{64}{\sqrt{3}}}\right)\right)= \end{equation} \begin{equation} 2 \coth ^{-1}\left(\frac{9}{\sqrt{81-\frac{64}{\sqrt{3}}}}\right), \end{equation}

So, is a "parallel" simplification possible in the original $\mbox{Li}_{2}$ case? (Of course, one might ask--more generally--about $\mbox{Li}_{n}$.)