Conjecture on boundary signs of Neumann eigenfunctions on Platonic solids

Let $u$ be a Neumann eigenfunction on a Platonic solid. Boundary-sign conjecture. Neumann eigenfunctions that are nonnegative on the boundary of a tetrahedron or octahedron do not exist, whereas Neumann eigenfunctions that are positive on the boundary of a dodecahedron or icosahedron do exist.

The conjecture concerns the existence of boundary-positive or boundary-nonnegative Neumann modes on the remaining Platonic solids. It is based on careful finite-element computations, and the source gives no resolution beyond this numerical evidence.