Path-cordiality characterization for finite abelian groups

Let $A$ be a finite abelian group, and let $\mathbb{P}$ be the class of path graphs. A group is $\mathbb{P}$-cordial if every path graph is $A$-cordial, where $A$-cordiality means that the induced vertex-label and edge-label partitions are almost rectangular. Path-cordiality conjecture. The group $A$ is $\mathbb{P}$-cordial if and only if $A$ is not a nontrivial product of copies of $\mathbb{Z}_2$; equivalently, if and only if there exists $a\in A$ with $|a|>2$. The conjecture is verified for all finite abelian groups of order less than $24$, and its odd-order direction is proved, while the even-order case with an element of order greater than $2$ remains partially open.