Tărnăuceanu's sum-of-element-orders conjecture for finite abelian groups

Let $G$ and $H$ be finite abelian groups of the same order. Define the sum-of-element-orders function by

$$\psi(G)=\sum_{g\in G}o(g),$$

where $o(g)$ denotes the order of $g$. Tărnăuceanu's conjecture. The groups $G$ and $H$ are isomorphic if and only if $\psi(G)=\psi(H)$.

This extends the corresponding result for finite abelian $p$-groups to all finite abelian groups. It asserts that, among finite abelian groups of a fixed order, the sum of element orders determines the isomorphism type; the supplied source does not indicate that the conjecture has been resolved.