The half-degree matching-removability conjecture

For $k\ge 1$, a vertex set $S$ in a $k$-connected graph $G$ is $k$-removable when $G-S$ remains $k$-connected; a matching is $k$-removable when its edge deletion leaves a $k$-connected graph. The half-degree matching-removability conjecture. Every $k$-connected graph $G$ with $\delta(G)\ge k+1$ contains a $k$-removable $\lceil(\delta(G)+1)/2\rceil$-matching, unless $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$. The paper's theorem supports this for $k\le 3$, except for the case $k=3$ and $\delta(G)=4$; the general conjecture remains open.