Regularity conjecture for the CW-complex of tilted Richardson cells

For $u,v\in S_n$, let $\mathcal{T}_{u,v}^{\geq0}$ be stratified by the positive tilted Richardson cells as

$$\mathcal{T}_{u,v}^{\geq0}=\bigsqcup_{[x,y]\subseteq[u,v]}\mathcal{T}_{x,y}^{>0}.$$

The source proves that this stratification forms a CW-complex.

Regularity conjecture for tilted Richardson cells. The CW-complex in the preceding stratification is regular.

Regularity would strengthen the established CW-complex structure and, together with the closure conjecture, would clarify the topology and face-poset structure of the nonnegative tilted Richardson variety. The source does not state that this conjecture is resolved.