The tree-width conjecture for Markov width

Let $G$ be a graph, and let $mu(G)$ denote its Markov width. The tree width of $G$ is the usual graph-theoretic tree-width.

Tree-width conjecture. The invariant $mu(G)$ is a function only of the tree width of $G$.

This conjecture proposes that graphs with the same tree width have the same Markov width. The source gives it as a conjectural extension suggested by the computed data.