The $0_k$-graph modular chromatic-index conjecture

Let $k$ be a positive integer. A $0_k$-graph is the graph class defined in the source, and $\chi'_k(G)$ is the mod $k$ chromatic index. $0_k$-graph conjecture. For every $0_k$-graph $G$,

$$\chi'_k(G)\leq k+o(k).$$

The source proposes this conjecture as a possible route to reducing the multiplicative constants in its main upper bounds; it would yield a $6k+o(k)$ bound for odd $k$ and an $8k+o(k)$ bound for even $k$.