The six-color conjecture for recoloring planar graphs of girth at least five

Let $G$ be a planar graph of girth at least five on $n$ vertices. The graph $R_6(G)$ is the reconfiguration graph whose vertices are proper $6$-colorings of $G$, with edges joining colorings that differ on one vertex.

Six-color conjecture. $R_6(G)$ has diameter $O(n)$.

Planar graphs of girth at least five are colorable from lists of size three, and the paper explains that seven colors already suffice by a simpler argument. The conjectured bound improves this to six colors; no resolution is given in the supplied text.