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As a corollary of our first main result, we verify the conjecture for abelian groups of the form $C_{p^m}\\oplus C_p, C_{p^m}\\oplus C_q, C_{p^m}\\oplus C_q^2$, $C_{p^m}\\oplus C_r^n$ where $p,q$ are distinct primes and $r\\in\\{2,3\\}$. 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