By [Fedor Petrov's answer](https://mathoverflow.net/a/457489) and [Korselt's criterion](https://en.wikipedia.org/wiki/Carmichael_number#Korselt's_criterion), the distinct primes $p_1,\dotsc,p_k$ provide a counterexample to the OP's conjecture if and only if their product $n$ is a Carmichael number. For example, the triple $p_1=3$, $p_2=11$, $p_3=17$ provides a counterexample, because their product $n=561$ is a Carmichael number.