Polynomial recoloring diameter conjecture for planar graphs

Let $G$ be a planar graph and let $k$ be an integer. The $k$-recoloring graph $R_k(G)$ has vertices corresponding to proper $k$-colorings of $G$, with edges joining colorings that differ on one vertex by one color. Polynomial recoloring diameter conjecture. For every planar graph $G$ and every integer $k\geq 7$, the graph $R_k(G)$ has polynomial diameter. This would establish a polynomial recoloring bound for planar graphs with at least seven colors; the statement is presented as a question for further work, and no resolution is given here.