Shephard's dimension-preserving subpolytope conjecture

A stacked polytope is a polytope obtained by repeatedly stacking simplices. A subpolytope is the convex hull of a subset of the vertices of a polytope. A combinatorial type records the face-incidence structure.

Shephard's conjecture. For every $d\ge 0$, every combinatorial type of $d$-dimensional polytope can be realized using subpolytopes of $d$-dimensional stacked polytopes.

The conjecture is the combinatorial form of Shephard's question after his geometric version was shown to be false; the source gives no resolution.