We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.
Benton, Alex and O'Rourke, Joseph, "A Class of Convex Polyhedra with Few Edge Unfoldings" (2008). Computer Science: Faculty Publications. 41.