Refinement of the perfect-power repulsion conjecture for partition numbers

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

$$M_k(d):=\max\left\{n:\Delta_k(n)\leq d\right\}.$$

Refinement of the perfect-power repulsion conjecture. If $k>1$, then $M_k(0)=1$, and for every $\varepsilon>0$,

$$M_k(d)=o\left(d^{\varepsilon}\right).$$

The first assertion recapitulates Sun's conjecture, while the second gives a sub-polynomial upper bound on the last partition index within distance $d$ of a $k$th power. It is presented as a numerical refinement and remains open.