The spanning conjecture for short paths in polytopes

Let $P$ be a $(d,2d-1)$-polytope with graph $G_P$, and let $x,y$ be vertices of $P$. A collection of $(x,y)$ paths spans the graph if removing one vertex from every path disconnects the graph. Spanning conjecture. The set of $(x,y)$ paths of $P$ of length $d-1$ or $d$ spans the polytope for every pair $(x,y)$. This is presented as a sufficient statement for the $d$-step conjecture, but the source gives no resolution.