Wormald's isomorphic linear-forest partition conjecture for cubic graphs

Let $G$ be a cubic graph with an even number of edges. Equivalently, $|V(G)|$ is divisible by $4$. A linear forest is a forest whose components are paths, and a linear partition is a partition of the edge set into linear forests.

Wormald's conjecture. There exists a linear partition of $G$ into two isomorphic linear forests.

This is equivalent to a $2$-edge-colouring whose monochromatic subgraphs are isomorphic linear forests. The paper reports no counterexamples below $32$ vertices, but the conjecture remains open.