Ji–Zhang–Zhao inequalities for overpartition ranks

Let $\overline N(a,c,n)$ denote the number of overpartitions of $n$ whose rank is congruent to $a$ modulo $c$.

Ji–Zhang–Zhao inequalities. For $n\geq 0$ and $1\leq i\leq 4$,

$$\overline N(0,10,5n+i)+\overline N(1,10,5n+i)\geq \overline N(4,10,5n+i)+\overline N(5,10,5n+i).$$

For $n\geq 0$,

$$\overline N(1,10,n)+\overline N(2,10,n)\geq \overline N(3,10,n)+\overline N(4,10,n).$$

These inequalities concern the relative sizes of residue classes of overpartition ranks. The paper states them as conjectures of Ji, Zhang and Zhao and Wei and Zhang; its abstract says that the inequalities are proved for $n=6$ and $n=10$, while the general assertions remain open in the supplied text.