Regular bipartite switch-connectivity threshold conjecture

Let $G$ be a $d$-regular balanced bipartite graph on $2n$ vertices, and let $f4d8^{\mathbf{connect}}_{k,\mathrm{reg}}(n)$ denote the minimum-degree threshold for the $k$-switch graph of such graphs to be connected. Regular switch-connectivity threshold conjecture. There exists $c\geq 0$ such that, for all $n$ and $k$,

$$\delta^{\mathbf{connect}}_{k,\mathrm{reg}}(n)\in\left[\frac{2n}{k+1}-c,\frac{2n}{k+1}+c\right].$$

The conjecture predicts that the connectivity threshold is within an additive constant of the degree of the cycle blow-up construction. The source notes that the upper bound is confirmed for $k=2$ and $k=3$, while the lower bound is known when $k+1$ divides $n$.