  For any graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated
edges, and a positive integer $r$, by $\chi'_{\Sigma,r}(G)$ we denote the
$r$-distant sum distinguishing index of $G$. This is the least integer $k$ for
which a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ exists such that
$\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every pair of distinct vertices
$u,v$ at distance at most $r$ in $G$. It was conjectured that
$\chi'_{\Sigma,r}(G)\leq (1+o(1))\Delta^{r-1}$ for every $r\geq 3$. Thus far it
has been in particular proved that $\chi'_{\Sigma,r}(G)\leq 6\Delta^{r-1}$ if
$r\geq 4$. Combining probabilistic and constructive approach, we show that this
can be improved to $\chi'_{\Sigma,r}(G)\leq (4+o(1))\Delta^{r-1}$ if the
minimum degree of $G$ equals at least $\ln^8\Delta$.
