Dichotomy for minimizers with at most three times the dimension in vertices

Determine whether the dichotomy between minimizers having either d+2 or d+3 facets extends to minimizers of the number of k-faces among d-dimensional polytopes with at most 3d-1 vertices.

Background

Lower-bound theorems for d-polytopes with 2d+1 and 2d+2 vertices exhibit a dichotomy among minimizers: the minimizing polytopes have either d+2 or d+3 facets. The paper studies an analogous refinement for d-polytopes with at most 2d vertices, specifically addressing the case of polytopes with at least d+3 facets, but it does not resolve whether the same structural dichotomy persists for the broader class of d-polytopes with at most 3d-1 vertices.

The unresolved question concerns the structure of k-face minimizers across a substantially larger vertex range. Establishing the conjectured extension would clarify whether the two facet counts d+2 and d+3 continue to govern the combinatorial types of minimizers beyond the ranges treated by the existing lower-bound theorems.