Distant sum distinguishing index of graphs

Consider a positive integer $r$ and a graph $G=(V,E)$ with maximum degree $\mathrm{\Delta }$ and without isolated edges. The least $k$ so that a proper edge colouring $c:E\to \{1,2,\dots ,k\}$ exists such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$foreverypairofdistinctvertices$u,v$ at distance at most $r$ in $G$ is denoted by ${\chi }_{\mathrm{\Sigma },r}^{\mathrm{\prime }}(G)$. For $r=1$ it has been proved that ${\chi }_{\mathrm{\Sigma },1}^{\mathrm{\prime }}(G)=(1+o(1))\mathrm{\Delta }$. For any $r\ge 2$ in turn an infinite family of graphs is known with ${\chi }_{\mathrm{\Sigma },r}^{\mathrm{\prime }}(G)=\mathrm{\Omega }({\mathrm{\Delta }}^{r-1})$. We prove that on the other hand, ${\chi }_{\mathrm{\Sigma },r}^{\mathrm{\prime }}(G)=O({\mathrm{\Delta }}^{r-1})$ for $r\ge 2$. In particular we show that ${\chi }_{\mathrm{\Sigma },r}^{\mathrm{\prime }}(G)\le 6{\mathrm{\Delta }}^{r-1}$ if $r\ge 4$.