Positroid stratification regular CW-complex conjecture

Let $Gr^{\geq 0}(k,n)$ be the nonnegative Grassmannian, with its positroid cell decomposition

$$Gr^{\geq 0}(k,n)=\bigsqcup_{\mathcal{M}\text{ is a positroid}}\Pi_{\mathcal{M}}.$$

For a positroid $\mathcal{M}$, let $\Pi_{\mathcal{M}}$ denote the corresponding positroid cell.

Positroid stratification conjecture. The positroid stratification of $Gr^{\geq 0}(k,n)$ is a regular CW-complex. In particular, the closure $\overline{\Pi}_{\mathcal{M}}$ of each positroid cell in $Gr^{\geq 0}(k,n)$ is homeomorphic to a closed ball.

The preceding result establishes that positroid cells are homeomorphic to open balls and that their decomposition is a CW-complex. The conjecture asks for the stronger regularity property, equivalently the closed-ball behavior of every cell closure.