No, for $n = 11$ this fails:

> 363 = 3 * 11^2 with [7, 2, 8, 5, 3, 4, 6, 9, 10, 1, 11]

> 484 = 2^2 * 11^2 with [10, 9, 6, 3, 1, 2, 4, 5, 7, 8,
> 11]


Running the code I wrote to check this a little more, there is more than one multiple of $n^2$ in the set you describe for all $11\leq n \leq 50$, see [here][1] for two permutations leading to different multiples of $n^2$ for each such $n$.


  [1]: https://mo271.github.io/mo/397577/non-unique.txt