On the smallest simultaneous power nonresidue modulo a prime

Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$n<p^{1/4 - c_r+o(1)}\quad(p\to\infty)$$ for some positive $c_r$ satisfying $c_r\ge e^{-(1+o(1))r} \; (r\to \infty).$