  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 $\mathsf c_0(G)$ 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 $\mathsf c_0(G)$ was first studied by Olson
and Peng in 1980's, 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 Gao et al. (Acta Arith. 168
(2015) 247-267) that $\mathsf c_0(G)=m(G)$. We confirm the conjecture for the
case when $G=C_{n_1}\oplus C_{n_2}$ with $n_1|n_2$, $n_1\geq 2p$, $p\geq 3$ and
$n_1n_2\geq 72p^6$.
