The dimension conjecture for the product-measure locus in the permutation simplex

Let $S_m$ be the symmetric group, let $\Delta(S_m)$ be the simplex of probability distributions on $S_m$, and let $P_m\subseteq\Delta(S_m)$ be the locus hit by product measures. A pure cycle is a permutation with exactly one nontrivial cycle and all other points fixed. Write

$$C(m)=\sum_{k=2}^m \frac{m!}{k\cdot (m-k)!}$$

for the number of pure cycles in $S_m$.

Dimension conjecture. For all $m$, the dimension of $P_m$ is exactly $C(m)$.

The paper proves the upper bound $\dim P_m\leq C(m)$ and verifies equality for $m\leq 7$, but the equality is conjectural for general $m$.