Total-coloring extension conjecture for matchings

Let $G$ be a graph with maximum degree $\Delta$, and let $H$ be a subgraph of $G$. A matching is a subgraph consisting of vertex-disjoint copies of $K_2$. A total-$k$-coloring of $H$ in $G$ is a total-$k$-coloring that is proper with respect to $G$.

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 $G$.

The conjecture asks whether the number of preassigned colors can be reduced from the general greedy bound $2\Delta+1$ to $\Delta+3$ without restrictions on the ambient graph or the matching. The paper proves the assertion for planar graphs when $\Delta\geq 28$, and notes that $\Delta+3$ is best possible in general.