No-cut set conjecture for the metric polytope

Let $\mathrm{met}_n$ be the metric polytope, and let its fractional vertices be the vertices that are not integral. The restriction of $\mathrm{met}_n$ to its fractional vertices is the graph whose vertices are these fractional vertices and whose edges are the edges of $\mathrm{met}_n$ joining them. No-cut set conjecture. For $n\geq 6$, the restriction of $\mathrm{met}_n$ to its fractional vertices is connected.

This conjecture asks whether the fractional vertices form a connected subgraph after the integral cut vertices are removed. The supplied text does not state a resolution, so its status remains open here.