Abreu–Nigro's homeomorphism conjecture for regular semisimple Lusztig varieties

Let $G=\operatorname{GL}_n(\mathbb{C})$, let $\mathbf{s}\in G^{\mathrm{rs}}$ be regular semisimple, and let $w,w'\in S_n$ be smooth permutations associated, via the natural combinatorial map from smooth permutations to linear Hessenberg subspaces, to the same linear Hessenberg space $H$. Thus $w$ is the unique codominant permutation with

$$X_H(s)=Y_w(\mathbf{s}),$$

and the varieties $Y_w(\mathbf{s})$ and $Y_{w'}(\mathbf{s})$ have isomorphic graded $S_n$-representations in cohomology. Abreu–Nigro's homeomorphism conjecture. The regular semisimple Lusztig varieties $Y_w(\mathbf{s})$ and $Y_{w'}(\mathbf{s})$ are homeomorphic. This would strengthen the known equality of their GKM graphs and graded $S_n$-representations by identifying the varieties topologically; the source gives no resolution of the conjecture.