Strong Ando conjecture for cubic graphs

Let $G$ be a cubic graph. A linear forest is a forest whose components are paths, and a bisection is a $2$-colouring of the vertex set with equally sized colour classes.

Strong Ando conjecture. Every cubic graph admits a bisection such that the two induced subgraphs are isomorphic linear forests.

This strengthens Ando's conjecture by requiring the common induced graph to be a linear forest. The paper notes that all known examples satisfy a bounded-component variant, with the Petersen graph requiring bound $4$, but the full conjecture remains open.