Three-weight edge-weighting problem for nice graphs

Let $G$ be a nice graph, meaning a graph without isolated edges, and let $a,b,c$ be pairwise distinct real numbers. A proper edge weighting is an edge weighting for which adjacent vertices have distinct weighted degrees. Three-weight edge-weighting problem for nice graphs. Does every nice graph admit an $\{a,b,c\}$-edge weighting for arbitrary given pairwise distinct real numbers $a,b,c$? The source presents this as a question lying between the 1-2-3 conjecture and the $(1,3)$-choosability conjecture, with no resolution supplied.