2 problems

Edwards et al.'s conjecture. If $H$ is a distance-2 matching, then every $(\Delta+1)$-edge-coloring of $H$ extends to a $(\Delta+1)$-edge-coloring of $G$.

Total-coloring extension conjecture for matchings. If $H$ is a matching, then every total-$(\Delta+3)$-coloring of $H$ in $G$ extends to a total-$(\Delta+3)$-coloring of $G$.