Merca–Ono–Tsai stabilization conjecture for partition numbers

For each fixed integer $d\geq 0$, define

$$L(d):=\max\{n:p(n)\leq d+1\},$$

and let $M_k(d)$ be the largest $n$ for which $p(n)$ is within $d$ of a $k$th power. Merca–Ono–Tsai's stabilization conjecture. There is a positive integer $N_d$ such that for all $k\geq N_d$,

$$M_k(d)=L(d).$$

The source attributes this to Merca–Ono–Tsai and gives no proof or resolution.