The vertex-facet product conjecture for 2-level polytopes

Let $P$ be a $2$-level $d$-polytope, and let $f_0(P)$ and $f_{d-1}(P)$ denote its numbers of vertices and facets, respectively. Vertex-facet product conjecture. For every $2$-level $d$-polytope $P$,

$$f_0(P)f_{d-1}(P)\leq d2^{d+1}.$$

Experiments establish the bound up to dimension $7$, and it is known for several infinite classes of $2$-level polytopes; the conjecture remains open in general.