Total sequence list 1-2-3 Conjecture

Let $G$ be any graph, and fix an ordering of the edges and vertices used in a total weighting. A total sequence $k$-list-weighting assigns to each edge and vertex a weight from its independently assigned list of $k$ real numbers; the induced vertex sequence records the relevant incident edge and vertex weights in the prescribed order. Let $\operatorname{ch}_\sigma^t(G)$ be the least $k$ such that every such list assignment permits adjacent vertices to receive different sequences. The total sequence list 1-2-3 Conjecture. For any graph $G$,

$$\operatorname{ch}_\sigma^t(G)\leq 2.$$

This is the total sequence version of the list 1-2-3 problem. The surrounding discussion gives a general upper bound of $3$, while the stated bound $2$ is left as a conjecture.