Asymptotic conjecture for near-perfect powers of overpartition numbers

Let $\overline{p}(n)$ denote the overpartition function, and define $M_{\overline{p},k}(d)$ analogously to $M_k(d)$ as the largest index whose overpartition number lies within $d$ of a $k$th power. The overpartition asymptotic conjecture. For $k\geq 2$,

$$M_{\overline{p},k}(d)\sim\frac{1}{\pi^2}\left(\frac{k}{k-1}\right)^2(\log d)^2$$

as $d\to\infty$. This is motivated by computational data and remains unproved.