The Hessenberg variety homeomorphism conjecture for smooth permutations

Let $SL_n(\mathbb{C})$ be the special linear group, let $X\in SL_n(\mathbb{C})$ be regular semisimple, and let $w\in S_n$ be smooth. Denote by $\mathcal{Y}_w(X)$ the variety associated to $w$ and $X$ in the paper, and call a permutation codominant according to the paper's definition.

Hessenberg variety homeomorphism conjecture. There exists a codominant permutation $w'$ such that

$$\mathcal{Y}_w(X)\text{ and }\mathcal{Y}_{w'}(X)\text{ are homeomorphic}.$$

The paper describes this as a strengthening of an earlier theorem, but the supplied text gives no resolution.