Polynomial combinatorial-type conjecture for shortest paths on convex polyhedra

Let $S$ be the boundary of a convex polyhedron, and let a shortest path $\gamma$ in $S$ have both endpoints in the interiors of facets. Its combinatorial type is the facet sequence $\mathcal L_\gamma$ traversed by $\gamma$.

Polynomial combinatorial-type conjecture. The cardinality of the set of combinatorial types of shortest paths in $S$ is polynomial in the number of facets of $S$ when the dimension is fixed.

This is stronger than the source-image conjecture because it does not fix one endpoint at a source point. The degree is expected to increase with the dimension, and the conjecture is known when the boundary has dimension $2$; the general case remains open.