I feel like it's true as for small cases I couldn't find counterexample.

In general, whether it's true that if we have prime number, $p_{1}, p_{2}, ...., p_{k}$ and $n=p_{1}p_{2}p_{3}...p_{k}$ then at least for one $ i \in $ {$1, 2, ...., k$} , $p^{i}$ divides $1^{(n-1)} +2^{(n-1)} + ...... + (p_{i}-1) ^{(n-1)}$? Each prime is greater than or equal to 3.