Vertex/facet trade-off conjecture for 2-level polytopes

Let $P$ be a $d$-dimensional 2-level polytope, meaning that every facet-defining hyperplane of $P$ has all vertices of $P$ on at most two parallel hyperplanes. Write $f_0(P)$ for the number of vertices and $f_{d-1}(P)$ for the number of facets.

Vertex/facet trade-off conjecture. One has

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

Moreover, equality is achieved if and only if $P$ is affinely isomorphic to the cross-polytope or the cube.

This conjecture seeks a sharp bound on the product of the numbers of vertices and facets of a $d$-dimensional 2-level polytope. The bound is supported by experimental results through dimension $7$, but the source provides no resolution in general.