  Consider a simple graph $G=(V,E)$ of maximum degree $\Delta$ and its proper
total colouring $c$ with the elements of the set $\{1,2,\ldots,k\}$. The
colouring $c$ is said to be \emph{neighbour sum distinguishing} if for every
pair of adjacent vertices $u$, $v$, we have $c(u)+\sum_{e\ni u}c(e)\neq
c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ for which it exists is denoted
by $\chi"_{\sum}(G)$, hence $\chi"_{\sum}(G) \geq \Delta+1$. On the other hand,
it has been daringly conjectured that just one more label than presumed in the
famous Total Colouring Conjecture suffices to construct such total colouring
$c$, i.e., that $\chi"_{\sum}(G) \leq \Delta+3$ for all graphs. We support this
inequality by proving its asymptotic version, $\chi"_{\sum}(G) \leq
(1+o(1))\Delta$. The major part of the construction confirming this relays on a
random assignment of colours, where the choice for every edge is biased by so
called attractors, randomly assigned to the vertices, and the probabilistic
result of Molloy and Reed on the Total Colouring Conjecture itself.
