Chan and Mao's monotonicity conjecture for overpartition ranks

Let $\overline{N}(m,n)$ denote the number of overpartitions of $n$ with $D$-rank $m$, and let $\overline{N2}(m,n)$ denote the number of overpartitions of $n$ with $M_2$-rank $m$. Chan and Mao's conjecture. For $(m,n)\neq (0,4)$ with $n\neq |m|+2$, one has

$$\overline{N}(m,n)\geq \overline{N}(m,n-1).$$

For $m\in\mathbb{Z}$ and $n\geq 0$, one has

$$\overline{N2}(m,n)\geq \overline{N2}(m,n-1).$$

These inequalities assert monotonicity in the size parameter for the two overpartition rank distributions, with the stated exception and restriction for the $D$-rank inequality. The conjecture was proposed by Chan and Mao in 2014; its resolution status is not established by the supplied source context.