Oscillation conjecture for the Hardy–Littlewood prime-counting inequality

Let $x\geq 2$ be large, let $r>0$ be real, and set $y=\log^r x$. With $\pi(t)=\#\{p\leq t:p\text{ is prime}\}$, the oscillation conjecture. Both inequalities

$$\pi(x+\log^r x)<\pi(x)+\pi(\log^r x)$$

and

$$\pi(x+\log^r x)>\pi(x)+\pi(\log^r x)$$

occur infinitely often as $x\to\infty$. This describes the proposed limitations of the Hardy–Littlewood inequality in short intervals; the supplied text gives no resolution of this assertion.