The 1-2 Conjecture for total weightings

Let $G=(V,E)$ be any graph. A total $k$-weighting is a function $w:V\cup E\to\{1,2,\ldots,k\}$, and the colour of $v\in V(G)$ is

$$w(v)+\sum_{e\ni v}w(e).$$

The 1-2 Conjecture. Every graph has a total $2$-weighting whose resulting vertex sums form a proper colouring; equivalently,

$$\chi_\Sigma^t(G)\leq 2.$$

The source says this conjecture remains open, while Kalkowski proved the weaker existence of a proper total weighting with vertex weights in $\{1,2\}$ and edge weights in $\{1,2,3\}$.