Summary: Let \(G\) be a finite abelian group and \(p\) be the smallest prime dividing \(|G|\). Let \(S\) be a sequence over \(G\). We say that \(S\) is regular if for every proper subgroup \(H\subsetneq G, S\) contains at most \(|H| -1\) terms from \(H\). Let \(c_0(G)\) be the smallest integer \(t\) such that every regular sequence \(S\) over \(G\) of length \(|(S| \geq t\) forms an additive basis of \(G\), i.e., \(\sum (S) = G\). The invariant \(c_0(G)\) was first studied by Olson and Peng in the 1980s, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than \(10^8\). In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by W. D. Gao et al. [Acta Arith. 168, No. 3, 247–267 (2015; Zbl 1330.11064)] that \(c_0(G) = m(G)\). We confirm the conjecture for the case when \(G = C_{n_1} \oplus C_{n_2}\) with \(n_1 \mid n_2, n_1 \ge 2p, p\ge 3\) and \(n_1n_2 \ge 72p^6\).