The $r$-adjacent strong chromatic index conjecture

For a graph $G$, two distinct vertices are $r$-adjacent if their distance is at most $r$. Let $\chi'_{a,r}(G)$ denote the least number of colours in a proper edge colouring of $G$ such that the sets of colours incident with any two $r$-adjacent vertices are distinct. Write $\Delta(G)$ and $\delta(G)$ for the maximum and minimum degrees of $G$, respectively.

The $r$-adjacent strong chromatic index conjecture. For each positive integer $r$ there exist constants $\delta_0$ and $C$ such that

$$\chi'_{a,r}(G)\leq \Delta(G)+C$$

for every graph without an isolated edge and with $\delta(G)\geq\delta_0$.

The conjecture proposes a uniform additive bound above the maximum degree when the minimum degree is sufficiently large. The source explains that the threshold $\delta_0$ should be roughly equal to $r$, up to a small additive constant, but does not state a resolution.