Ando's colouring conjecture for cubic graphs

Let $G$ be a cubic graph. A $2$-vertex colouring is a map

$$c_V:V(G)\longrightarrow\{B,W\}.$$

The monochromatic induced subgraphs are the subgraphs induced by the vertices of each colour.

Ando's colouring conjecture. A cubic graph $G$ admits a $2$-vertex colouring $c_V$ such that the monochromatic induced subgraphs are isomorphic.

This is a colouring reformulation of Ando's bisection conjecture; isomorphic monochromatic subgraphs automatically have equally many vertices. The conjecture remains open.