The k-optimal set edge-colouring conjecture

Let $G$ be a graph, let $k\in\mathbb{Z}^+$, and let $D$ be a $k$-optimal set, meaning a $k$-dependent set maximizing $\phi_k(D)=k|D|-|E(G[D])|$. A subgraph is $k$-edge-chromatic if its edges can be coloured with $k$ colours so that adjacent edges receive different colours. The $k$-optimal set edge-colouring conjecture. If $D$ is a $k$-optimal set in $G$, then $G$ has a $k$-edge-chromatic subgraph in which every vertex of $V(G)-D$ has degree $k$.

This conjecture is introduced as a natural strengthening of the results developed earlier and is stated to imply the special case of Tuza's conjecture for graphs of the form $I_k\vee H$. Its resolution is not supplied in the source.