The convex-polytope realization question for convex codes

A code is a collection of subsets of a finite ground set. A code is convex if it has a realization by convex sets, and it is realized with convex polytopes if those realizing sets can be chosen to be convex polytopes.

Convex-polytope realization question. Can every convex code be realized with convex polytopes?

An affirmative answer would strengthen the paper's characterization of polytope-convex codes and could help determine whether convex-code recognition is algorithmically decidable. The source presents this as an open question; its resolution would be a first step toward resolving the convex code decision problem.