Is $|\pi(x) - Li(x)| < \frac{\sqrt{\ln(x) x}}{8 \pi}$ for $x>q$. for some fixed $q$ still potentially consistant with RH?

Question

Let $\pi(x)$ be the prime counting function.

From the question

If $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x) $ is true, what does that say about the Riemann zeta zero's?

came the related question :

Is

$|\pi(x) - Li(x)| < \frac{\sqrt{\ln(x) x}}{8 \pi}$ for $x>q$.

for some fixed $q$ still potentially consistant with RH ?

And if so, what does that say about the nontrivial zero's ?

I felt I had to split up that question in two hence this question.