Vertex-Unfolding Polyhedra

Consider a polyhedron with simply connected facets (no holes on a facet) and without boundary (every edge is incident to exactly two facets). Can the polyhedron be cut along potentially all of its edges, but leaving certain faces connected at vertices, and unfolded into one piece in the plane without overlap? Such an unfolding is called a vertex-unfolding, to distinguish from widely studied edge-unfoldings (see Problem 9) and general unfoldings. An important subproblem here is whether all convex polyhedra have vertex-unfoldings; a negative answer would also resolve Problem 9.