The $0_{q(k)}$-graph modular chromatic-index conjecture

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

$$\chi'_k(G)\leq 3k+o(k).$$

The source presents this as a second proposed route to improving the multiplicative constants in the main theorem. The supplied context does not state a resolution or the precise definition of $q(k)$ and the associated graph class.