Mihail–Vazirani conjecture on edge-expansion of 0/1 polytopes

Let $P$ be a $0/1$ polytope, meaning the convex hull of a subset of the vertices of a hypercube, and let $G_P$ be its graph (the graph whose vertices are the vertices of $P$ and whose edges are its one-dimensional faces). Write $h(G_P)$ for the edge-expansion of $G_P$.

Mihail–Vazirani conjecture. Every $0/1$ polytope $P$ satisfies

$$h(G_P) \ge 1.$$

High-dimensional hypercubes have edge-expansion $1$ by Harper's inequality, and this conjecture asserts that arbitrary $0/1$ polytopes expand at least as well. Its resolution status is not specified in the supplied text.