Asymptotic upper-bound conjecture for distant irregularity strength

Let $r\geq 2$ be an integer, and let $G$ be a graph with maximum degree $\Delta$ and without an isolated edge. Write $s_r(G)$ for the least number of colours needed in an edge-colouring whose induced vertex sums distinguish vertices at distance at most $r$. The asymptotic distant-irregularity conjecture. For every integer $r\geq 2$ and each such graph $G$,

$$s_r(G)\leq (1+o(1))\Delta^{r-1}.$$

Here the asymptotic term is with respect to the relevant growth of $\Delta$. The conjecture is presented as an asymptotically optimal upper bound, while the supplied text gives no resolution. It concerns improving the second-order terms in the paper's bounds, particularly for graphs with relatively large minimum degree.