Guo–Zeng's strengthened truncated Gauss inequality for overpartitions

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. For $n,k\geq 1$, Guo–Zeng's conjecture.

$$(-1)^{k-1}\left(\overline{p}(n)+2\sum_{j=1}^k(-1)^j\overline{p}(n-j^2)\right)+\overline{p}(n-k^2)\geq 0,$$

with strict inequality if $n\geq k^2$. The conjecture was proved independently by Mao in 2015 and Yee in 2015, and later reconfirmed by Wang and Yee in 2019.