Mutation connectivity conjecture for uniform matroid polytopes of rank 4

A uniform matroid polytope of rank $4$ is a uniform matroid polytope with rank $4$. Consider two such matroid polytopes on the same ground set, and require every intermediate object in a mutation sequence to remain a matroid polytope.

Mutation connectivity conjecture for uniform matroid polytopes. Every pair of uniform matroid polytopes of rank $4$ on the same ground set can be transformed into each other by a finite number of mutations, preserving the matroid polytope property.

This problem is motivated by the Cordovil–Las Vergnas conjecture, which asserts mutation connectivity for uniform oriented matroids. The source presents the rank-$4$ matroid-polytope version as an open problem.