Surjectivity conjecture for homomorphisms between projective cubes

Let $r$ and $k$ be integers with $r\geq k$, and let $\mathcal{PC}_{2r}$ and $\mathcal{PC}_{2k}$ denote projective cubes. A graph homomorphism is a map preserving adjacency. Surjectivity conjecture. Every homomorphism from $\mathcal{PC}_{2r}$ to $\mathcal{PC}_{2k}$ must be onto. This is presented as an equivalent formulation of the surjectivity conjecture for non-bipartite binary Cayley graphs; the paper does not establish it in general, though it proves a special case.