Shephard's Conjecture

Shephard's conjecture states that every convex polyhedron admits a self-unoverlapping unfolding (Shephard 1975). This question is still unsettled (Malkevitch), though most mathematicians believe that the answer is yes.

It is known that Shephard's conjecture is false for non-convex 3-dimensional polyhedra (Bern et al. 1999, Malkevitch).