Weak cordiality conjecture for trees and abelian groups

A family of graphs is weakly $A$-cordial if all but finitely many of its elements are $A$-cordial, where $A$ is an abelian group. Weak cordiality conjecture. For any abelian group $A$, the set of trees is weakly $A$-cordial. The result is known for $A=\mathbb{Z}_2^2$, since all trees except $P_4$ and $P_5$ are $\mathbb{Z}_2^2$-cordial. The conjecture remains open for arbitrary abelian groups.