There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected "edge" unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of L).
@article{arxiv.0806.2360,
title = {Existence of a polyhedron which does not have a non-overlapping pseudo-edge unfolding},
author = {Alexey S Tarasov},
journal= {arXiv preprint arXiv:0806.2360},
year = {2008}
}