Conjecture on the positive growth exponent of the extremal-prime counting function

Let $\mathbb E$ be the set of extremal prime numbers and define its counting function by

$$\pi_{\epsilon}(x)=\sum_{p\in\mathbb E,\ p\leq x}1.$$

Positive-exponent conjecture. There exists an infimum

$$\inf\left\{\alpha>0:\pi_{\epsilon}(x)=o(x^{\alpha})\right\}$$

and it is positive. The paper reports that numerical data place the corresponding exponent near $\gamma/2$, where $\gamma$ is the Euler constant; the asserted positive value remains unproved.