I am considering a combinatorial argument which involves the following quantity. We use the prime counting function $\pi(n)$ and to save on exponents we set $h=\pi(n/2)$. The quantity as a function of integer $n \gt 7$ is $$(\pi(n)!)^{1/(n-h)}$$

Computations for small $n$ suggest this is always less than $4$, as do rough back-of-the-envelope asymptotic calculations. Is this bounded above for all $n \gt 7$? If so, what is the bound? (I'm hoping it is always less than 3.)

Gerhard "Researching Minds Want To Know" Paseman, 2020.05.30.