The homeomorphism conjecture for the Springer variety and the crossingless-matching space

Let $S=\mathbb{S}^2$ and let $B$ be the set of crossingless matchings on $2n$ points. For each $a\text{ in }B$, let $S_a$ be the submanifold of $S^{\mathord{\times}2n}$ consisting of tuples whose coordinates agree on every pair in $a$, and set

$$\widetilde{S}=\bigcup_{a\in B}S_a.$$

Let $\mathcal{B}_{n,n}$ be the $(n,n)$ Springer variety. The homeomorphism conjecture. The spaces $\mathcal{B}_{n,n}$ and $\widetilde{S}$ are homeomorphic. This would strengthen the already established agreement of their component and intersection structures and explain their isomorphic cohomology rings; the source gives no resolution of the conjecture.