General-graph isolated-matching threshold conjecture

Let $\delta^{\mathbf{no\ iso}}_k(n)$ denote the minimum-degree threshold such that the $k$-switch graph of an $n$-vertex graph, when nonempty, is guaranteed to have positive minimum degree. Isolated-matching threshold conjecture. There exists $c>0$ such that, for all $k$ and $n$,

$$\delta^{\mathbf{no\ iso}}_k(n)\in[n/k-c,n/k+c].$$

The conjecture is motivated by the semidegree version of the Caccetta–Häggkvist conjecture, which is equivalent to the corresponding no-isolated-vertices threshold for balanced bipartite graphs. The general-graph threshold remains open.