Monotonicity conjecture for the diameter of Schrijver graphs

Let $k$ and $r$ be integers with $r\geq 1$, and let $D(G)$ denote the diameter of a graph $G$. For $n=2k+r$, write $\operatorname{SG}(n,k)$ for the Schrijver graph. Monotonicity conjecture. If $r\geq 1$, then

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

The authors have determined the exact diameter when $r\leq 2$ and when $r\geq k-3$, and provide bounds in the remaining range. Their computations suggest that the diameter is non-increasing as $r$ increases, but the conjecture remains open.