The triangle-free cubic graph conjecture for the Clebsch graph

Let $G$ be a cubic graph, meaning every vertex has degree $3$. Let $PQ_4$ denote the four-dimensional projective cube, also known as the Clebsch graph. A graph homomorphism $G\to PQ_4$ is a vertex map preserving adjacency.

Triangle-free cubic graph conjecture. Every triangle-free cubic graph is homomorphic to $PQ_4$.

A graph containing a triangle cannot map to the triangle-free Clebsch graph, so triangle-freeness is the natural obstruction highlighted in the paper. The paper proves the assertion for cubic graphs of sufficiently large girth, but leaves the full triangle-free case unresolved.