Is it known wether the following limit tends to Infinity or not? Is there any possibility for it to converge to a constant?

$$\lim_{n \to \infty} \sum^n_{p\le n} \frac{1}{\sqrt{p}} - li(\sqrt{n})$$

(Being $p$ a prime number and $li(k)$ the logarithmic integral)

I have not found anything about it. It seems that most efforts are made trying to upper-bound the asymptotic behaviour of the limit. For example, here there is a proof of this behaviour based on the Prime Number Theorem. Also, the Riemann's Hypothesis provides a tighter upper bound. But what about a lower bound?