Regular CW decomposition conjecture for the momentum amplituhedron

Let $\mathcal{M}_{n,k}$ be the momentum amplituhedron, and let $\mathcal{S}_{n,k}$ consist of the decorated permutations whose strata are boundary strata, together with the decorated permutation $\sigma_{k,n}$. For each $\sigma\in\mathcal{S}_{n,k}$, let $\Phi^\circ_{\sigma}$ be the corresponding stratum. Regular CW decomposition conjecture. There is a regular CW decomposition

$$\mathcal{M}_{n,k}=\bigsqcup_{\sigma\in\mathcal{S}_{n,k}}\Phi^\circ_{\sigma}$$

of the momentum amplituhedron. In particular, each boundary stratum $\Phi^\circ_{\sigma}$ is homeomorphic to an open ball. Boundary stratifications had been synthesized computationally for $4\leq n\leq 8$ and $2\leq k\leq n-2$, motivating the conjecture, but the source gives no general proof.