Diagram matroid basis-count conjecture

Let $w\in S_n$, and let $DM_w$ be the diagram matroid of $w$, obtained from a generic matrix $[A\mid I_n]$ whose entries satisfy $A_{ij}=0$ whenever $(i,j)\in D(w)$, where

$$D(w)=\{(i,w(j))\in[n]\times[n]:i<j,\ w(i)>w(j)\}.$$

For a permutation $v$, let $\ell_1,\ldots,\ell_k$ be the lengths of the runs of anti-fixed points of $v$, and write $C_m$ for the $m$th Catalan number. Diagram matroid basis-count conjecture. For any $w\in S_n$, the number of bases of $DM_w$ is

$$\sum_{\substack{v\geq w\text{$v$ avoids $123$}}}C_{\ell_1+1}\cdots C_{\ell_k+1}.$$

This extends the corresponding basis-count formula to diagram matroids. The source indicates that a proof might be obtained by Möbius inversion, but that a more substantial sign-reversing involution is needed; the conjecture is not reported as resolved.