Tutte-polynomial equality for diagram matroids and positroids

Let $w\in S_n$. Let $DM_w$ be the diagram matroid associated to the permutation diagram of $w$, and let $\mathscr{P}_w$ be the corresponding positroid. Tutte-polynomial equality conjecture. For any $w\in S_n$, the Tutte polynomial of $DM_w$ equals the Tutte polynomial of $\mathscr{P}_w$. The equality has been checked through $S_7$, while the source does not report a proof in general.