The planar graph Markov width conjecture

Let $G$ be a planar graph, and let $mu(G)$ denote its Markov width.

Planar graph Markov width conjecture. There is a universal constant $C$ such that

$$\mu(G) \leq C$$

whenever $G$ is planar. Even more strongly, one may take $C=6$.

The conjecture is motivated by computations for small irreducible graphs and by the bounded-degree Markov bases known for cycles and complete bipartite graphs $K_{2,n}$.