Edge-criticality conjecture for the quadrangulation graph $QG(n,k)$

For integers $k\geq 1$ and $n\geq 2k+1$, let $QG(n,k)$ be the spanning subgraph of the Schrijver graph $SG(n,k)$ constructed in Theorem 1.1, which is a quadrangulation of $\mathbb{P}^{n-2k}$ and satisfies $\chi(QG(n,k))=n-2k+2$. A graph is edge-critical if deleting any edge decreases its chromatic number. Edge-criticality conjecture. For any $k\geq 1$ and $n\geq 2k+1$, $QG(n,k)$ is edge-critical. The graph $QG(n,k)$ has the same chromatic number as $SG(n,k)$, and this conjecture proposes the edge-critical analogue of Schrijver's vertex-criticality result; it remains open in the source.