  We consider the following extension of the concept of adjacent strong edge
colourings of graphs without isolated edges. Two distinct vertices which are at
distant at most $r$ in a graph are called $r$-adjacent. The least number of
colours in a proper edge colouring of a graph $G$ such that the sets of colours
met by any $r$-adjacent vertices in $G$ are distinct is called the $r$-adjacent
strong chromatic index of $G$ and denoted by $\chi'_{a,r}(G)$. It has been
conjectured that $\chi'_{a,1}(G)\leq\Delta+2$ if $G$ is connected of maximum
degree $\Delta$ and non-isomorphic to $C_5$, while Hatami proved that there is
a constant $C$, $C\leq 300$, such that $\chi'_{a,1}(G)\leq\Delta+C$ if
$\Delta>10^{20}$ [J. Combin. Theory Ser. B 95 (2005) 246--256]. We conjecture
that a similar statement should hold for any $r$, i.e., that for each positive
integer $r$ there exist constants $\delta_0$ and $C$ such that $\chi'_{a,r}(G)
\leq \Delta+C$ for every graph without an isolated edge and with minimum degree
$\delta \geq \delta_0$, and argue that a lower bound on $\delta$ is unavoidable
in such a case (for $r>2$). Using the probabilistic method we prove such upper
bound to hold for graphs with $\delta\geq \epsilon\Delta$, for every $r$ and
any fixed $\varepsilon\in(0,1]$, i.e., in particular for regular graphs. We
also support the conjecture by proving an upper bound $\chi'_{a,r}(G) \leq
(1+o(1))\Delta$ for graphs with $\delta\geq r+2$.
