Criticality conjecture for interlacing edges of large Schrijver graphs

Let $d$ be any integer, let $k_d$ be a threshold depending on $d$, and let ${\rm SG}(2k+d,k)$ be the Schrijver graph. An edge is interlacing according to the standard interlacing-edge definition for Schrijver graphs, and an edge is critical if deleting it lowers the chromatic number. Interlacing-edge criticality conjecture. For every $d$ and every $k>k_d$, all interlacing edges of ${\rm SG}(2k+d,k)$ are critical. This would characterize the critical edges for Schrijver graphs of large size with fixed small chromatic-number excess, extending the corresponding theorem cited in the source; the full characterization of critical edges is not known.