2 problems

Distance-3 matching extension conjecture. If $H$ is a distance-3 matching, then every total-$(\Delta+2)$-coloring of $H$ extends to a total-$(\Delta+2)$-coloring of $G$.

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 $G$.