Existence of shard polytopes for congruence-uniform posets of regions

Let $\mathcal{H}$ be a hyperplane arrangement and let $\mathsf{B}$ be a base region such that the poset of regions $\mathsf{PR}(\mathcal{H},\mathsf{B})$ is a congruence-uniform lattice. A shard polytope for a shard $\mathsf{S}$ is a polytope whose normal fan has walls containing $\mathsf{S}$ and contained in the union of shards forcing $\mathsf{S}$. Shard-polytope existence conjecture. Every shard admits a shard polytope. This stronger conjecture would imply polytopality of all quotient fans in the preceding conjecture via Minkowski sums; it is motivated by the known type $A$ and type $B$ constructions and remains open in general.