The facet-connectedness conjecture for cutsets

Let $P$ be a polytope, let $F$ be a facet of $P$, and let $C$ be a cutset. Facet-connectedness conjecture. The cutset $C$ cannot contain vertices $u$ and $v$ lying in a common facet of $F$ unless there exists a connected path from $u$ to $v$ within $C\cap F$. The source uses this claim in an argument about preserving path lengths under polytope perturbations, but gives no resolution.