Strengthening of the Schrijver graph quadrangulation theorem

Let $n$ and $k$ be integers with $n>2k\geq 2$, and let $SG(n,k)$ denote the Schrijver graph. A quadrangulation of $P^{n-2k}$ is a graph embedded in the projective space $P^{n-2k}$ so that every face is a quadrilateral; it is non-bipartite if its graph is not bipartite. A spanning subgraph contains all vertices of $SG(n,k)$. Schrijver graph quadrangulation conjecture. The Schrijver graph $SG(n,k)$ contains a non-bipartite quadrangulation of $P^{n-2k}$ as a spanning subgraph. This would strengthen the stated theorem, which constructs a non-bipartite quadrangulation of $P^{n-2k}$ admitting a homomorphism into $SG(n,k)$; whether the stronger spanning-subgraph assertion holds is left as an open question.