Eli's descent-set conjecture for zero-marked cyclic permutations

Let ${\mathcal T}^0_n$ be the set of $n$-cycles in one-line notation with one entry replaced by $0$, and let $D(a)=\{i:1\le i\le n-1,\ a_i>a_{i+1}\}$ denote the descent set of a sequence $a$. For $I\subseteq[n-1]$, write $\{\tau\in {\mathcal T}^0_n:D(\tau)=I\}$ for the zero-marked cyclic permutations with descent set $I$.

Eli's descent-set conjecture. For any $n$ and any $I\subseteq[n-1]$,

$$\left|\{\tau\in {\mathcal T}^0_n\,:\,D(\tau)=I\}\right|=\left|\{\sigma\in\mathcal S_n\,:\,D(\sigma)=I\}\right|.$$

This conjecture asserts that the descent-set distribution on zero-marked cyclic permutations agrees with that on all permutations. The supplied text attributes it to Eli; its resolution is not stated in the provided context, so its status is recorded as open.