6-color conjecture for cubic bipartite planar graph squares

Let $G$ be a cubic bipartite planar graph, and let $G^2$ be its square, obtained by joining vertices at distance at most $2$ in $G$. Let $\chi(G^2)$ denote the chromatic number of $G^2$.

Cubic bipartite planar square-coloring conjecture. The square satisfies

$$\chi(G^2)\leq 6.$$

This is a strengthening of the general cubic planar bound in Wegner's conjecture and is presented as an open problem for cubic bipartite planar graphs.