  Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total
colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices
$u,v\in V$ are sum distinguished if they differ with respect to sums of their
incident colours, i.e. $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The
least integer $k$ admitting such colouring $c$ under which every $u,v\in V$ at
distance $1\leq d(u,v)\leq r$ in $G$ are sum distinguished is denoted by ${\rm
ts}_r(G)$. Such graph invariants link the concept of the total vertex
irregularity strength of graphs with so called 1-2-Conjecture, whose concern is
the case of $r=1$. Within this paper we combine probabilistic approach with
purely combinatorial one in order to prove that ${\rm ts}_r(G)\leq
(2+o(1))\Delta^{r-1}$ for every integer $r\geq 2$ and each graph $G$, thus
improving the previously best result: ${\rm ts}_r(G)\leq 3\Delta^{r-1}$.
