Equidistribution conjecture for ascents and descents of perfect matchings

Let $\PM_{2n}$ denote the set of perfect matchings on $[2n]$, and let $\Asc(m)$ and $\Des(m)$ be the ascent and descent sets of a matching $m\in\PM_{2n}$, defined by the relative order of $s_i\cdot m$ and $m$ in the matching order. For $J\subseteq[2n-1]$, write

$$\boldsymbol{t}^J:=\prod_{j\in J}t_j.$$

Equidistribution conjecture. For all $n\in\mathbb{N}$,

$$\sum_{m\in\PM_{2n}}\boldsymbol{t}^{\Asc(m)}=\sum_{m\in\PM_{2n}}\boldsymbol{t}^{\Des(m)}.$$

This asserts that the ascent and descent set statistics on perfect matchings are equidistributed. The equality has been verified experimentally for $2n\leq14$; the stronger question of Schur-positivity with respect to $\Des$ remains unresolved.