Here $\mu(n)$ is Möbius function and $M(x)$ is Mertens function.

The computations show that the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$ stays between $-0.2$ and $-1.2$ when $10^1<x<10^8$, and it has been shown herethat the series diverges.

I understand that Gonek's conjecture about how $M(x)/\sqrt{x}$ grows. The computations show that $M(x)/\sqrt{x}$ stays between $-0.5$ and $+0.5$ in the range $10^3<x<10^8$. Mertens - now disproved - conjecture, stated that $M(x)/\sqrt{x}$ would stay between $-1$ and $+1$, but for a very large number - may be as high as $~10^{10^{39}}$ - the bounds would be broken.

Is there a conjecture about the upper and lower bounds enveloping the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$? (Here the bounds may be in form of function of x or constants).