Erdős's conjecture on primes of prescribed multiplicative order

For an integer $a$ and a positive integer $r$, define

$$E_a(r)=\#\{p\text{ prime}:\operatorname{ord}_p(a)=r\},$$

where $\operatorname{ord}_p(a)$ is the multiplicative order of $a$ modulo the prime $p$. Erdős's conjecture. For every $\epsilon>0$,

$$E_a(r)\ll_\epsilon r^\epsilon.$$

The conjecture is introduced as a second well-known hypothesis used to control the number of prime factors of $a^n-1$; the source gives no resolution status.