Figueiredo et al.'s conjecture on chordal graph edge-coloring

Let $G$ be a chordal graph, meaning a graph with no induced cycle of length at least $4$. Let $\Delta(G)$ be its maximum degree. A graph is Class 2 if its edges cannot be properly colored with $\Delta(G)$ colors, and it is subgraph-overfull if it has a subgraph $H$ satisfying $|V(H)|$ odd and

$$|E(H)|>\Delta(G)\frac{|V(H)|-1}{2}.$$

Figueiredo et al.'s conjecture. $G$ is Class 2 if and only if it is subgraph-overfull. This conjecture proposes a characterization of edge-chromatic Class 2 chordal graphs by the subgraph-overfull obstruction. The supplied text gives no resolution beyond noting that the paper's results cover a class of split graphs.