Minimum-degree conjecture for minimal extendable claw-free graphs

Let $G$ be a minimal $k$-extendable claw-free graph, meaning that $G$ is $k$-extendable and deleting any edge produces a graph that is not $k$-extendable. Let $\delta(G)$ denote its minimum degree. Minimum-degree conjecture. The minimum degree of $G$ is either

$$2k$$

or

$$2k+1.$$

The paper proves that the minimum degree of a minimal $3$-extendable claw-free graph is either $6$ or $7$, and proposes the displayed assertion for general $k$ as a direction for further investigation.