Ando's bisection conjecture for cubic graphs

Let $G$ be a cubic graph. A bisection is a $2$-colouring of the vertex set with colour classes of equal size, and the monochromatic induced subgraphs are the subgraphs induced by the two colour classes.

Ando's conjecture. Every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic.

This conjecture is related to both the Ban–Linial and Wormald conjectures and remains open.