Do perfect powers repel partition numbers?

In 2013 Zhi-Wei Sun conjectured that $p(n)$ is never a power of an integer when $n>1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If $k>1$ and ${\mathrm{\Delta }}_k(n)$ is the distance between $p(n)$ and the nearest $k$th power, then for every $d\ge 0$ we conjecture that there are at most finitely many $n$ for which ${\mathrm{\Delta }}_k(n)\le d\mathrm{.}$ More precisely, for every $\epsilon >0,$ we conjecture that $$M_k(d):=\max \{n:{\mathrm{\Delta }}_k(n)\le d\}=o(d^{\epsilon })\mathrm{.}$$ In $k$-power aspect with $d$ fixed, we also conjecture that if $k$ is sufficiently large, then $$M_k(d)=\max \{n:p(n)-1\le d\}\mathrm{.}$$ In other words, $1$ generally appears to be the closest $k$th power among the partition numbers.