Conjecture on bounded-conflict antimagic labellings

Let $G$ be a finite, simple, undirected graph. A graph is $C$-antimagic if it admits an edge labelling whose vertex sums have at most $C$ conflicts, as defined in the paper. $C$-antimagic conjecture. Every connected graph except $K_2$ is $C$-antimagic, for some constant $C\geq 0$. This conjecture proposes a bounded-conflict weakening of antimagicness. The paper introduces $C$-antimagicness as a form of weak antimagicness; whether every connected graph other than $K_2$ satisfies the property for some constant remains open.