The 1-2-Conjecture for total graph weightings

Let $G$ be a graph. A total weighting is a function $\omega:E(G)\cup V(G)\to\{1,2\}$, and its induced total vertex weight is

$$w(v):=\omega(v)+\sum_{u\in N(v)}\omega(uv).$$

The 1-2-Conjecture. For every graph $G$, there is a weighting $\omega:E(G)\cup V(G)\to\{1,2\}$ such that the induced total vertex weights properly color $V(G)$.

This is the total version of the 1-2-3 conjecture, formulated by Przybyło and Woźniak. The source notes that Kalkowski came close to settling it by allowing edge weights from $\{1,2,3\}$ and vertex weights from $\{1,2\}$, but the two-valued total-weighting conjecture remains open.