Unit-gap conjecture for diameters of Schrijver graphs

Let $k$ and $r$ be integers with $2\leq r\leq k-2$, and let $D(G)$ denote the diameter of a graph $G$. For $n=2k+r$, write $\operatorname{SG}(n,k)$ for the Schrijver graph. Unit-gap conjecture. If $2\leq r\leq k-2$, then

$$D\left(\operatorname{SG}(2k+r,k)\right)-D\left(\operatorname{SG}(2k+r+1,k)\right)\in\\{0,1\\}.$$

The conjecture concerns the possibility of larger drops in the diameter as $r$ increases. It is based on computations and is open; the paper notes that the analogous differences are in $\\{0,1\\}$ for $k\leq 5$, while a larger gap occurs at the transition from $r=1$ to $r=2$ when $k\geq 6$.