Merca–Ono–Tsai finiteness conjecture for near-perfect powers of partition numbers

For fixed integers $k>1$ and $d\geq 0$, define the distance from $p(n)$ to the closest $k$th power by

$$\Delta_k(n):=\min\left\{|p(n)-m^k|:m\in\mathbb{Z}\right\}.$$

Merca–Ono–Tsai's finiteness conjecture. For fixed integers $k>1$ and $d\geq 0$, there are at most finitely many $n$ for which $\Delta_k(n)\leq d$. This finiteness assertion is used to define the maximal such $n$, but no resolution is given in the source.