Favaron–Shi minimum-degree conjecture for minimal factor-critical graphs

Let $G$ be a $k$-factor-critical graph, meaning that deleting any $k$ vertices leaves a graph with a perfect matching. Call $G$ minimal if deleting any edge produces a graph that is not $k$-factor-critical. Write $(G)$ for the minimum vertex degree of $G$.

Favaron–Shi's conjecture. If $G$ is a minimal $k$-factor-critical graph, then

$$\delta(G)=k+1.$$

The conjecture was posed by Favaron and Shi and concerns the sharpness of the general lower bound on the minimum degree of minimal $k$-factor-critical graphs. It is known for several values of $k$, including $k=1$, $k\in\{n-6,n-4,n-2\}$, and $k\in\{2,n-10,n-8\}$, as well as for claw-free graphs; the source proves it for planar graphs.