Wei–Zhang inequalities for overpartition ranks modulo 6

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

Wei–Zhang inequalities. For $n\geq 11$,

$$\overline N(0,6,3n)\geq \overline N(1,6,3n)=\overline N(3,6,3n)\geq \overline N(2,6,3n),$$ $$\overline N(0,6,3n+1)\geq \overline N(1,6,3n+1)=\overline N(3,6,3n+1)\geq \overline N(2,6,3n+1),$$

and

$$\overline N(1,6,3n+2)\geq \overline N(2,6,3n+2)\geq \overline N(0,6,3n+2)\geq \overline N(3,6,3n+2).$$

These are inequalities among overpartition rank classes in residue classes modulo $6$. The supplied text attributes them to Wei and Zhang and gives no resolution evidence, so their general status is left open.