> ***Answer:*** *No this is not true.*

For $m=5$, a counter example is: $(0, 2, 0, 2, 0, 2, 1, 2, 1, 1, 1, 0, 1, 2, 0)$. In this case we have:
$$\begin{align}
U(1) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 4 & 4 & 3 & 3 & 2 & 1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0
\end{array}\right) \\
U(2) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 0 & 0 & 4 & 4 & 5 & 5 & 4 & 4 & 3 & 2 & 1 & 1 & 0 & 0 \\
1 & 2 & 1 & 3 & 2 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0
\end{array}\right) \\
U(3) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 4 & 4 & 5 & 5 & 5 & 5 & 4 & 4 & 3 & 2 & 1 & 1 & 0 & 0 \\
2 & 4 & 3 & 4 & 3 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 1
\end{array}\right) \\
U(4) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
4 & 5 & 5 & 5 & 5 & 5 & 5 & 4 & 4 & 3 & 2 & 1 & 1 & 0 & 0 \\
4 & 5 & 4 & 4 & 3 & 4 & 3 & 3 & 2 & 2 & 2 & 2 & 3 & 3 & 2
\end{array}\right) \\
U(5) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
5 & 5 & 5 & 5 & 5 & 5 & 5 & 4 & 4 & 3 & 2 & 1 & 5 & 4 & 4 \\
5 & 5 & 4 & 5 & 4 & 5 & 4 & 4 & 3 & 3 & 3 & 3 & 5 & 5 & 4
\end{array}\right) 
\end{align}$$

Each $U(k)$ has only $14$ entries $\geq k$. 

To elaborate a bit: 

 - For $m\leq4$ the statement in your question is true, as can be checked by complete enumeration of all cases.
 - For $m=5$ if you mod out cyclic permutations there are precisely 20 counter examples:
$$\{(1, 1, 1, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 1, 2), (1, 1, 1, 1, 0, 2, 0, 2,
0, 0, 0, 2, 2, 1, 2), (1, 1, 1, 1, 0, 2, 0, 0, 2, 0, 2, 0, 2, 1, 2), (1,
1, 1, 0, 1, 2, 0, 0, 2, 0, 2, 0, 2, 1, 2), (1, 1, 1, 0, 2, 1, 0, 2, 0,
0, 2, 0, 2, 1, 2), (1, 1, 1, 0, 2, 1, 0, 0, 2, 0, 2, 0, 2, 1, 2), (1, 1,
1, 0, 0, 2, 0, 2, 0, 2, 1, 2, 0, 1, 2), (1, 1, 1, 0, 0, 2, 0, 2, 0, 2,
1, 0, 2, 1, 2), (1, 1, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 2), (1, 1, 1,
0, 0, 2, 0, 0, 2, 2, 0, 2, 1, 1, 2), (1, 1, 0, 1, 2, 2, 1, 2, 2, 2, 0,
0, 1, 0, 0), (1, 1, 2, 1, 2, 2, 2, 2, 0, 1, 0, 1, 0, 0, 0), (1, 0, 1, 0,
1, 2, 1, 0, 2, 1, 2, 2, 2, 0, 0), (1, 0, 1, 0, 1, 0, 1, 2, 2, 1, 2, 2,
2, 0, 0), (1, 0, 1, 0, 1, 2, 1, 2, 2, 2, 0, 1, 2, 0, 0), (1, 0, 1, 0, 1,
2, 1, 2, 2, 2, 2, 0, 1, 0, 0), (1, 0, 1, 0, 1, 2, 1, 2, 2, 2, 0, 2, 1,
0, 0), (1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 2, 2, 2, 0, 0), (1, 0, 1, 2, 1, 2,
2, 2, 0, 1, 2, 0, 1, 0, 0), (1, 0, 1, 2, 1, 2, 2, 2, 2, 0, 1, 0, 1, 0,
0)\}$$
 - I would expect that there counter examples for all $m\geq5$.