Doikou’s conjecture on commutativity of shuffle elements

Let $H_n(q)$ be the generic Iwahori--Hecke algebra of the symmetric group $d504_n$ with standard basis $T_w$ indexed by $w\in \mathfrak{S}_n$. For $1\leq i\leq \frac{n(n-1)}{2}$, define the shuffle element $\mathscr{Y}_{n,i}=\sum_{\substack{w\in\mathfrak{S}_n\ell(w)=i}}T_w$. Doikou's conjecture asserts that, for every $n\geq 2$ and all admissible $i,j$, the shuffle elements commute: $[\mathscr{Y}_{n,i},\mathscr{Y}_{n,j}]=0$; equivalently, $\mathscr{Y}_{n,i}\mathscr{Y}_{n,j}=\mathscr{Y}_{n,j}\mathscr{Y}_{n,i}$ whenever $1\leq i,j\leq \frac{n(n-1)}{2}$.