A while ago, I asked a similar question For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true? where users mathworker21 and WE Tutorial School proved that for non-square $k$, $n\le k$ is always true when $n!+k$ is sqaure.
Recently I got the idea to check if the property is true for all perfect powers and using PARI, I observed that if, $$n!+k=a^b$$ where $n, k, a, b\in \Bbb{N}$, $k\gt 3$ and $b\ge 2$, then $n\le k$ is always true. In search for a counter-example, I covered a range of $k\le 2500$ and $n\le 10^4$ for each $k$ and found none.
So my question is,
If true, can we prove/partially prove that $n\le k$ always holds? If false, then what is the smallest counter-example where $n\gt k$?