Proof of a conjecture on congruences of the sum of minimal excludants

Very recently, Baruah et al. refined a result of Andrews and Newman on the sum of minimal excludants over all the partitions of a positive integer n by establishing the generating functions of two functions [Formula: see text] and [Formula: see text] which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively. They proved some congruences on [Formula: see text] and [Formula: see text] modulo 4 and 8 and posed a conjecture on [Formula: see text] and [Formula: see text] at the end of their paper. In this paper, we confirm this conjecture by using theta function identities.