Eulerian descent-set distribution conjecture for marked cycles

For $n\geq 1$, let ${\mathcal T}^0_n$ be the class of objects defined in the paper, let $D(\sigma)\subseteq\{1,2,\dots,n-1\}$ denote the descent set of $\sigma\in{\mathcal T}^0_n$, let ${\mathcal S}_n$ be the symmetric group, and let $A_n(x)$ be the $n$-th Eulerian polynomial.

Eulerian descent-set distribution conjecture. For any $n$ and any subset $D\subseteq\{1,2,\dots,n-1\}$,

$$|\{\sigma\in{\mathcal T}^0_n\,:\,D(\sigma)=D\}|=|\{\pi\in{\mathcal S}_n\,:\,D(\pi)=D\}|.$$

In particular,

$$\sum_{\sigma\in{\mathcal T}^0_n}x^{\operatorname{des}(\sigma)+1}=A_n(x).$$

The conjecture arose from experimental evidence and was checked by computer for $n$ up to $9$; the asserted distribution beyond those computations remains open.