Proof of a conjecture on a congruence modulo 243 for overpartitions

Let \(\bar{p}(n)\) denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding \(\bar{p}(n)\). In particular, Xia discovered several infinite families of congruences modulo 9 and 27 for \(\bar{p}(n)\). Moreover, Xia conjectured that for \(n\ge 0\), \(\bar{p}(96n+76) \equiv 0\ (\mathrm{mod}\ 243)\). In this paper, we confirm this conjecture by using theta function identities and the (p, k)-parametrization of theta functions.