Botler, Colucci and Kohayakawa's additive bound conjecture for the mod $k$ chromatic index

Let $G$ be a simple graph and let $k\geq 2$ be an integer. A $\chi'_{k}$-coloring of $G$ is an edge coloring such that the subgraph induced by the edges of each color has all degrees congruent to $1\pmod{k}$; write $\chi'_{k}(G)$ for the minimum number of colors in such a coloring. Botler, Colucci and Kohayakawa's conjecture. There is a constant $C$ such that

$$\chi'_{k}(G)\leq k+C$$

for every graph $G$. This conjecture asks whether the linear bound can always have additive constant independent of $k$; the paper improves the previously known bound from $198k-101$ to $177k-93$, but does not resolve the conjecture.