Young-type inequality conjecture for the prime-counting function

Let $\pi(t)$ denote the number of primes at most $t$, and let $p,q>1$ satisfy

$$\frac{1}{p}+\frac{1}{q}=1.$$

There exists a sufficiently large natural number $n_0$ such that, for all $x,y\geq n_0$, Young-type conjecture.

$$\pi(xy)\leq\frac{\pi(x^p)}{p}+\frac{\pi(x^q)}{q}.$$

The source presents this as the first of three hypotheses motivated by analogies with Young, Hölder, and Minkowski inequalities; its resolution is not specified.