The $d$-step conjecture for simple polytopes

Let $P$ be a simple $d$-polytope with $2d$ facets, and let $u$ and $v$ be two complementary vertices, meaning vertices not lying in a common facet. The $d$-step conjecture. There is a path of length $d$ from $u$ to $v$.

This is the crucial $n=2d$ case underlying the Hirsch and non-revisiting conjectures. Its status is not established by the supplied source context.