Existence of a homogeneous mono-unstable convex polyhedron with seven vertices

Determine whether there exists a homogeneous convex polyhedron in three dimensions that has exactly one unstable equilibrium point and exactly seven vertices. Resolving this would close the gap left by the proof that no such polyhedron exists with at most six vertices and would establish whether the lower bound of seven vertices is tight.

Background

The paper proves that every homogeneous mono-unstable convex polyhedron must have at least seven vertices, by reducing the problem to families of quadratic inequality systems and certifying their infeasibility via semidefinite programming for all cases with five or six vertices. For seven vertices, their method fails to certify infeasibility for a small subset of cases, leaving open whether such a polyhedron can exist.

In the Discussion, the authors enumerate four possibilities to explain the inconclusive outcome for seven vertices, and explicitly express a conjecture that existence at seven vertices is highly unlikely. This frames a precise existence question: does a homogeneous mono-unstable polyhedron with exactly seven vertices exist?