Mao's rank and $M_2$-rank inequalities modulo 6 and 10

Let $N(s,m,n)$ denote the number of partitions of $n$ whose Dyson rank is congruent to $s$ modulo $m$. For partitions with no repeated odd parts, define the $M_2$ rank of a partition $\lambda$ by

$$\left\lceil \frac{l(\lambda)}{2} \right\rceil-n(\lambda),$$

and let $N_2(s,m,n)$ denote the number of such partitions whose $M_2$ rank is congruent to $s$ modulo $m$. The inequalities below are asserted for the indicated ranges of $n$: in the second, third, and seventh inequalities $n\geq 1$, and in the others $n\geq 0$.

Mao's conjectured inequalities. Computation evidence suggests that

$$\begin{aligned} N(0,10,5n)+N(1,10,5n)&>N(4,10,5n)+N(5,10,5n),\\ N(1,10,5n)+N(2,10,5n)&\geq N(3,10,5n)+N(4,10,5n),\\ N_2(0,10,5n)+N_2(1,10,5n)&>N_2(4,10,5n)+N_2(5,10,5n),\\ N_2(0,10,5n+4)+N_2(1,10,5n+4)&>N_2(4,10,5n+4)+N_2(5,10,5n+4),\\ N_2(1,10,5n)+N_2(2,10,5n)&>N_2(3,10,5n)+N_2(4,10,5n),\\ N_2(1,10,5n+2)+N_2(2,10,5n+2)&>N_2(3,10,5n+2)+N_2(4,10,5n+2),\\ N_2(0,6,3n+2)+N_2(1,6,3n+2)&>N_2(2,6,3n+2)+N_2(3,6,3n+2). \end{aligned}$$

The paper proves only a limited version of one of Mao's inequalities. The source states that the remaining three conjectures after other work are still open, so the full displayed collection is recorded as open.