Skowronek-Kaziόw's total product-colouring conjecture

Let $G$ be a graph, and let $\chi_\Pi^t(G)$ be the least $k$ for which a total weighting from $\{1,\ldots,k\}$ properly colours adjacent vertices by the products of the vertex weight and all incident edge weights.

Skowronek-Kaziόw's conjecture. For every graph $G$,

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

The conjecture is known for 3-colourable and complete graphs, while the source gives the general bound $\chi_\Pi^t(G)\leq 3$. Thus the proposed bound remains open.