nt.number theory – Is $prod_{k=1}^nk^{sigma(k)}$ a square or a cube for some $sigmain S_n$?

Note that for any permutation $sigmain S_5$ the product $prod_{k=1}^5k^{sigma(k)}$ is neither a square nor a cube.

Question. Let $n>5$ be an integer. Is the product $prod_{k=1}^nk^{sigma(k)}$ a square for some $sigmain S_n$? Is the product $prod_{k=1}^nk^{sigma(k)}$ a cube for some $sigmain S_n$?

The question looks not very challenging, and I believe that the answer should be positive. Any ideas to provide a proof?

The question can be refined further, for example, I conjecture that for any integer $n>5$ there is a permutation $sigmain S_n$ with $sigma(1)=1$ and $sigma(2)=2$ such that $prod_{k=1}^n k^{sigma(k)}=(p-1)^3$ for some prime $p$. Let $a(n)$ denote the number of such permutations $sigma$. I find that
$$a(6)=1, a(7)=3, a(8)=2, a(9)=27, a(10)=44, a(11)=154.$$

For the above question, I ask for an actual proof of the positive answer, rather than inaccurate heuristic arguments.