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Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) totalcolouring 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 theirincident colours, i.e. $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. Theleast integer $k$ admitting such colouring $c$ under which every $u,v\in V$ atdistance $1\leq d(u,v)\leq r$ in $G$ are sum distinguished is denoted by ${\rmts}_r(G)$. Such graph invariants link the concept of the total vertexirregularity strength of graphs with so called 1-2-Conjecture, whose concern isthe case of $r=1$. Within this paper we combine probabilistic approach withpurely 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 $G$, thusimproving the previously best result: ${\rm ts}_r(G)\leq 3\Delta^{r-1}$.

Distant total irregularity strength of graphs via random vertex ordering