  In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the
authors have extended the concept of quadrangulation of a surface to higher
dimension, and showed that every quadrangulation of the $n$-dimensional
projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite.
They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the
Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a
quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the
conjecture.
