Distant sum distinguishing index of graphs

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