Closure conjecture for positive tilted Richardson cells

Let $\mathcal{T}_{u,v}^{>0}$ be the totally positive part of a tilted Richardson variety, and let $\mathcal{T}_{u,v}^{\geq0}$ be its totally nonnegative part.

Positive-part closure conjecture. The closure of the positive part equals the nonnegative part:

$$\overline{\mathcal{T}_{u,v}^{>0}}=\mathcal{T}_{u,v}^{\geq0}.$$

If true, this would identify the closure of each positive cell with the corresponding nonnegative cell and would support a CW-complex structure whose face poset is the interval poset of the tilted Bruhat interval. The source notes that the classical proof strategy does not apply because tilted Bruhat orders lack a unique maximal element, and gives no resolution.