The list neighbour sum distinguishing total colouring problem

Let $G$ be a graph with maximum degree $\Delta$. In the natural list version of neighbour sum distinguishing total colouring, colours are chosen from arbitrary lists of real numbers, and the corresponding list parameter is defined by requiring a proper total colouring whose weighted degrees distinguish adjacent vertices. List version problem. The natural list correspondent of $\chi”_{\sum}(G)$ is bounded from above by

$$(1+o(1))\Delta.$$

The source presents this as an interesting problem rather than an established result; its methods for the ordinary parameter do not apply to the list version.