  In 1980, Bressoud conjectured a combinatorial identity $A_j=B_j$ for $j=0$ or
$1$. In this paper, we introduce a new partition function $\overline{B}_0$
which can be viewed as an overpartition analogue of the partition function
$B_0$. An overpartition is a partition such that the last occurrence of a part
can be overlined. We build a bijection to get a relationship between
$\overline{B}_0$ and $B_1$, based on which an overpartition analogue of
Bressoud's conjecture for $j=0$ is obtained.
