Lovász's perfect-matching conjecture for cubic vertex transitive graphs

Let $\Gha$ be a finite cubic vertex transitive graph, and let $M$ be a perfect matching in $\Gha$. Write $\Gha \backslash M$ for the graph obtained by deleting the edges of $M$. Lovász's conjecture. There exists a perfect matching $M$ in $\Gha$ such that $\Gha \backslash M$ consists either of one cycle, in which case $\Gha$ is Hamiltonian, or of two disjoint cycles of the same length. If true, this would imply that all finite vertex transitive cubic graphs have a uniform partite presentation; the source describes the conjecture as still open.