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, $p_{1}, p_{2},\dotsc, p_{k}$ and $n=p_{1}p_{2}p_{3}\dotsb p_{k}$ then at least for one $ i \in\{1, 2, \dotsc, k\}$, $p_i$ divides $1^{n-1} +2^{n-1} + \dotsb + (p_{i}-1) ^{n-1}$? Each prime is greater than or equal to 3.