Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.

The naive heuristic that $r$-th roots of unity should be "randomly distributed" suggests that there should be infinitely many such primes, and indeed they should have density 1 among all primes. Can this be made rigorous?