Let G be an additive finite abelian group. Let S = g1 •. . .•g l be a sequence over G, and k(S) = ord(g1) −1 + • • • + ord(g l ) −1 be its cross number. Let t(G) (resp. η(G)) be the smallest integer t such that every sequence of t elements, repetition allowed, from G has a non-empty zero-sum subsequence when G is cyclic, and for any integer r ≥ 3 there are infinitely many groups G of rank r such that t(G) > η(G). Girard (2012) conjectured that t(G) = η(G) for all finite abelian groups of rank

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... This conjecture has been verified only for the groups G ∼ = Cpα ⊕ Cpα , G ∼ = C2 ⊕ C2p and G ∼ = C3 ⊕ C3p with p ≥ 5, where p is a prime. We confirm this conjecture for more groups, including the groups G ∼ = Cn ⊕ Cn with the smallest prime divisor of n not less than the number of distinct prime divisors of n.