PSPACE-completeness conjecture for 4-To-3 recoloring

Let $G$ be a graph. A $4$-To-$3$ instance asks whether a proper $4$-coloring of $G$ can be transformed into a proper $3$-coloring by repeatedly recoloring one vertex while maintaining a proper coloring. The 4-To-3 conjecture. $4$-To-$3$ is PSPACE-complete. This is a more specific proposed complexity result for graph-coloring reconfiguration; the source presents it after discussing how PSPACE-completeness for restricted mixing problems would transfer to larger numbers of colors, and gives no resolution of the conjecture.