Partition-theoretic stabilization conjecture for kth-power repulsion

Let $p(n)$ be the partition function. For $k>1$ and $d\geq 0$, define

$$\Delta_k(n):=\min\left\{\lvert p(n)-m^k\rvert:m\in\mathbb Z\right\},$$ $$M_k(d):=\max\left\{n:\Delta_k(n)\leq d\right\},$$

and

$$L(d):=\max\left\{n:p(n)-1\leq d\right\}.$$

Partition-theoretic stabilization conjecture. For each non-negative integer $d$, there is a positive integer $N_d$ such that for $k\geq N_d$,

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

Here $L(d)$ is described using the fact that $p(n)-1$ counts nonempty partitions of size at most $n$ without parts of size $1$. The conjecture strengthens stabilization by identifying the eventual value, reflecting the claim that $1$ is generally the closest $k$th power to the partition numbers. It is based on numerical evidence and remains open.