Pineda-Villavicencio's lower-bound conjecture for polytopes with $2d+\ell$ vertices

Let $d\geqslant 3$ and $1\leqslant\ell\leqslant d-1$. Let $P$ be a $d$-polytope with $2d+\ell$ vertices, and let $T(r)$ denote the $r$-simplex. Pineda-Villavicencio's conjecture.

1. If $P$ has at least $d+3$ facets, then

$$f_k(P)\geqslant f_k(J(\ell+1,d))$$

for each $k$ with $1\leqslant k\leqslant d-2$. 2. If $P$ has $d+2$ facets, then $P$ is a $(d-a)$-fold pyramid over $T(m)\times T(a-m)$ for some $2\leqslant a\leqslant d$ and $2\leqslant m\leqslant\floor{a/2}$.

The conjecture refines the proposed minimisers for $d$-polytopes with at most $3d-1$ vertices. The paper states that it resolves this conjecture in full; the cases $\ell=1$ and $\ell=2$ had already been settled, and every minimiser in part (i) is shown to have exactly $d+3$ facets.