The 1-2-3 conjecture for locally finite polygonal tilings

Let $\mathcal T$ be a locally finite tiling with a finite number of polygonal prototiles. Suppose

$$\mathcal T=\bigcupdot_{k=1}^N(\varLambda_k+\mathcal P_k)$$

is a partition into translations of finitely many, not necessarily distinct, finite subgraphs, and let

$$l_k:\mathcal P_k\to\{1,2,3\}$$

be edge-weightings. The 1-2-3 conjecture for locally finite polygonal tilings. There exist such a partition and weightings $l_k$ that produce a global solution to the 1-2-3 problem, meaning that the resulting weighted degrees of adjacent vertices are different. This conjecture gives a more nuanced patch-based formulation after purely locally derivable solutions are shown not to exist for some tilings; the source does not state whether it has been resolved.