By Fedor Petrov's answer, 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.
GH from MO
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