Thomassen's crumby coloring conjecture for 3-connected cubic graphs

A crumby coloring of a graph is a red-blue vertex coloring in which the blue subgraph has maximum degree at most $1$ and the red subgraph has minimum degree at least $1$ and contains no path with $3$ edges. Thomassen's conjecture. Every $3$-connected cubic graph has a crumby coloring. This conjecture was refuted by Bellitto, Klimošová, Merker, Witkowski and Yuditsky, who constructed an infinite family of $3$-connected cubic counterexamples; related open cases include outerplanar, $K_4$-minor-free and bipartite graph classes.