The edge-deletion monotonicity conjecture for Markov width

Let $G$ be a graph, let $H$ be obtained from $G$ by deleting an edge, and let $\mu(G)$ and $\mu(H)$ denote their Markov widths.

Edge-deletion monotonicity conjecture. If $H$ is obtained from $G$ by deleting an edge, then

$$\mu(H)\leq\mu(G).$$

The statement extends the proved minor monotonicity result for vertex deletion and edge contraction. The authors state that it agrees with their data but that they have been unable to prove it.