  Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to
conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of
$5k+4$ could be divided into sub-classes 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 \cite{atkin}, who proved Dyson's
conjecture. More recently, Mao proved several equalities and inequalities,
leaving some as conjectures, for rank differences for partitions modulo 10
\cite{mao10} and for $M_2$ rank differences for partitions with no repeated odd
parts modulo $6$ and $10$ \cite{maom2}. Alwaise et. al. proved four of Mao's
conjectured inequalities \cite{swisher}, while leaving three open. Here, we
prove a limited version of one of the inequalities conjectured by Mao.
