We showed that if $q^{p-1}\equiv 1 \pmod {p^2}$ for all sufficiently large primes $p$, then $q^{p-1}\equiv 1 \pmod {p^3}$ for infinitely many primes $p$.

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We proved the result in the title as a consequence of Faltings' Theorem. This was proved, more-or-less simultaneously, by Heath-Brown