Asymptotic conjecture for near-perfect powers of partition numbers

Let $\Delta_k(n)$ be the distance from $p(n)$ to the closest $k$th power, and let

$$M_k(d):=\max\{n\in\mathbb{N}:\Delta_k(n)\leq d\}.$$

The asymptotic conjecture for $M_k(d)$. For $k\geq 2$,

$$M_k(d)\sim\frac{3}{2\pi^2}\left(\frac{k}{k-1}\right)^2(\log d)^2$$

as $d\to\infty$. This is motivated by computational data and complements a proved lower bound, but remains unproved.