The short-interval prime number conjecture

Let $<\varepsilon<1$ and let $\Phi(x)=x^{\varepsilon}$, where $\Phi(x)$ determines the intervals $I(x,\Phi)=[x-\Phi(x),x+\Phi(x)]$. Write $\pi(I(x,\Phi))$ for the number of prime integers in this interval. Short-interval prime number conjecture. The asymptotic formula

$$\pi\big(I(x,\Phi)\big)\sim \frac{\#\big(I(x,\Phi)\big)}{\log x}$$

holds as $x\to\infty$. This is a classical conjecture about the density of primes in intervals of length $x^{\varepsilon}$; the source states that it remains open in full generality.