Kaplan–Lev–Roditty's antimagic conjecture for trees

Let $A$ be a finite Abelian group, and write $I(A)$ for the set of involutions of $A$. Set $A^*=A\setminus\{0\}$. An $A^*$-antimagic labeling of a graph with $|A|$ vertices is a bijection from its edges to $A^*$ such that all vertex sums are pairwise distinct. Kaplan–Lev–Roditty's antimagic conjecture. A tree with $|A|$ vertices is $A^*$-antimagic if and only if $|I(A)|\neq1$. The supplied source reports that this conjecture was proved for $4$-trees, but does not establish the full conjecture; the general assertion is therefore not resolved by the stated evidence.