Equivistal subdivision conjecture for convex polyhedra

Let $S$ be the boundary of a convex polyhedron. Two source points are equivistal if their vistal trees are isomorphic and corresponding nodes represent the same facet sequences.

Equivistal subdivision conjecture. The equivalence relation induced by equivistality constitutes a convex polyhedral subdivision of $S$. Moreover, the number of open regions in this subdivision is polynomial in the number of facets of $S$.

This conjecture gives a geometric strengthening of the expected polynomial control on vistal-tree types as the source point varies. It remains open in the generality stated.