EXTENSIVELY EDITED by David Speyer. The previous version was very confused, but Steven Sivek and Graham, in the comments, figured out what was going on.

G. C. Shephard, in his paper "Twenty Problems on Convex Polyhedra: Part I", associates to a three dimensional polyhedron the sequence $(p_3, p_4, p_5, p_6, \ldots)$, with $p_k$ being the number of facets that are $k$-gons. He poses the problem of characterizing all sequences of integers which arise in this way.

Are there any developments and references on this problem?