The simple-polytope lower-interval conjecture for the face lattice of a phase tropical hypersurface

Let $\hat n=\{0,\dots,n\}$, let $\sigma=\langle I_1,\dots,I_k\rangle$ be a cyclic partition of $\hat n$, and let $J\subseteq\hat n$. Write $(\sigma,J)\in\mathcal W$ when $J$ contains elements in at least two of the subsets $I_1,\dots,I_k$; in this case, $\sigma$ divides $J$. The partial order is $(\sigma',J')\preceq(\sigma,J)$ when $\sigma$ is a refinement of $\sigma'$ and $J'\subseteq J$, and define

$$\mathcal W_{\preceq(\sigma,J)}:=\{(\sigma',J')\in\mathcal W:(\sigma',J')\preceq(\sigma,J)\}.$$

Simple-polytope lower-interval conjecture. For each element $(\sigma,J)\in\mathcal W$, the lower interval $\mathcal W_{\preceq(\sigma,J)}$ is isomorphic to the face lattice of a simple polytope.

This conjecture would describe the local combinatorial structure of the regular CW-complex whose face lattice is $\mathcal W$. The source does not provide evidence of a resolution, so its status remains open.