The commutativity conjecture for excedance–antiexcedance elements

Let $S_n$ be the symmetric group, and let $\mathbf{k}[S_n]$ be its group algebra. For a permutation $w\in S_n$, write $\operatorname{exc}w$ for the number of $i\in[n]$ such that $w(i)>i$, and $\operatorname{anxc}w$ for the number of $i\in[n]$ such that $w(i)<i$. For $a,b\in\mathbb{N}$, define

\mathbf{X}_{a,b}:=\sumnonlimits\limits_{\substack{w\in S_n;\operatorname{exc}w=a;\operatorname{anxc}w=b}}w\in\mathbf{k}[S_n].

Commutativity conjecture. For fixed $n$, the elements $\mathbf{X}_{a,b}$ commute for all $a,b\in\mathbb{N}$; equivalently,

$$\mathbf{X}_{a,b}\mathbf{X}_{c,d}=\mathbf{X}_{c,d}\mathbf{X}_{a,b}$$

for all $a,b,c,d\in\mathbb{N}$. This is presented as an open problem concerning a family of combinatorially defined elements in the symmetric group algebra; the excerpt gives no evidence of a resolution.