Ballantine–Merca inequality for truncated overpartition sums

Let $\overline{M}_k(n)$ denote the number of overpartitions of $n$ in which the first part larger than $k$ appears at least $k+1$ times. For $n,k\geq 1$, Ballantine–Merca's conjecture.

$$\sum_{j=-\infty}^{\infty}(-1)^j\overline{M}_k\bigl(n-j(3j-1)\bigr)\geq 0,$$

with strict inequality if $n\geq (k+1)^2$. This conjecture was settled by Yao in 2025.