Bounded additive group irregularity strength for graphs without small components

Let $G$ be a graph of order $n$ with no connected components of order less than $3$. A $\Gamma$-irregular labeling is a labeling of the edges of $G$ by elements of an abelian group $\Gamma$ such that the induced vertex weights are pairwise distinct; denote the group irregularity strength by the least group order guaranteeing such a labeling for every abelian group of that order. Bounded additive group irregularity conjecture. There exists a constant $K$ such that every such graph $G$ has a $\Gamma$-irregular labeling for every group $\Gamma$ satisfying

$$|\Gamma|\geq n+K.$$

The conjecture asserts a uniform additive bound on the group irregularity strength for graphs whose components all have order at least $3$, extending the preceding bounds for broad classes of disconnected graphs. Its resolution would determine whether the dependence on the graph order can always be reduced to $n+O(1)$ in this setting.