Weak path-cordiality conjecture for finite abelian groups

Let $A$ be a finite abelian group, and let $\mathbb{P}$ be the class of path graphs. A group is weakly $\mathbb{P}$-cordial if all but finitely many path graphs are $A$-cordial. Weak path-cordiality conjecture. Every finite abelian group is weakly $\mathbb{P}$-cordial. This is motivated by examples showing weak $\mathbb{P}$-cordiality for products of two and three copies of $\mathbb{Z}_2$, together with the existence of infinitely many $A$-cordial paths for each $A$; the conjecture remains open in general.