Ramanujan’s congruence \(p(5k+4) \equiv 0 \pmod 5\) led Dyson (Eureka 8:10–15, 1944) to define a measure “rank”, and then conjectured that \(p(5k+4)\) partitions of \(5k+4\) could be divided into subclasses with equal cardinality to give a direct proof of Ramanujan’s congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer (Some properties of partitions 4:84–106, 1954), who proved Dyson’s conjecture. More recently, Mao (Number Theory 133:3678–3702, 2013) proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 and for \(M_2\)-rank differences for partitions with no repeated odd parts modulo 6 and 10 (Mao in Ramanujan J 37:391–419, 2015). Alwaise et al. (Ramanujan J. doi:10.1007/s11139-016-9789-x, 2016) proved four of Mao’s conjectured inequalities, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.