Resolve Smilansky’s conjecture on indecomposable polytopes without triangular faces

Determine whether an indecomposable polytope without triangular 2-faces exists in any dimension, thereby resolving Smilansky’s conjecture.

Background

The paper discusses Smilansky’s result that every indecomposable 3-dimensional polytope must have at least four triangular facets. It then states Smilansky’s broader conjecture that no indecomposable polytope without triangular 2-faces exists in any dimension.

The authors explicitly note that the conjecture remains unresolved and express doubt that it is true. Their accompanying lower bound on deformation-cone dimension in terms of the number of triangular facets gives additional restrictions in dimension three, but does not settle the conjecture in arbitrary dimensions.

References

A well-known result of Smilansky Corollary~6.8 states that any indecomposable $3$-polytope must have at least $4$ triangular facets, and Smilansky conjectured that there does not exist an indecomposable polytope without triangular $2$-faces in any dimension. This conjecture is still open, although we do not believe it to be true.

— Deformation cones of 2-cubical combinatorial zonotopes  (2610.13441 - Curtis et al., 9 Oct 2026) in Section 2, immediately after Lemma 2.?? (discussion of Smilansky’s result and Lemma \ref{lem:smil_app})