The 1-2 Conjecture for total product-labellings

Let $G$ be a graph. A total $k$-labelling is a map $\ell:V(G)\cup E(G)\to\{1,\dots,k\}$. For each vertex $u$, let $\pi_\ell(u)$ be the product of the labels incident to $u$, including the label of $u$. The labelling is product-proper when adjacent vertices have distinct products, and $\chi^t_{\rm P}(G)$ is the least $k$ for which such a total labelling exists. The 1-2 Conjecture (product version). For every graph $G$,

$$\chi^t_{\rm P}(G)\leq 2.$$

The conjecture was considered by Skowronek-Kaziόw; the paper states that it remains widely open, while a general upper bound of $3$ is known.