Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation $$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}. $$ Basic theorems about Dirichlet series imply that if the Dirichlet series on the right converges for some $s = \sigma + it$, then it converges for all $s$ with real part $> \sigma$. Hence, $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ converging is a sufficient condition for the Riemann hypothesis.

One way to approach this question is to use partial summation. Let $M(x) = \sum_{n \leq x} \mu(n)$. Then $$ \sum_{n \leq x} \frac{\mu(n)}{\sqrt{n}} = \frac{M(x)}{\sqrt{x}} + \frac{1}{2} \int_{1}^{x} \frac{M(t)}{t^{3/2}} \, dt. $$ Odlyzko and te Riele proved that $\liminf_{x \to \infty} \frac{M(x)}{\sqrt{x}} < -1.009$ and $\limsup_{x \to \infty} \frac{M(x)}{\sqrt{x}} > 1.06$. Much earlier, Ingham had showed that $M(x)/\sqrt{x}$ was unbounded assuming the linear independence of the imaginary parts of the zeroes of $\zeta(s)$.

In addition, Gonek has an unpublished conjecture (mentioned in Ng's paper "The distribution of the summatory function of the Mobius function") that $$ -\infty < \liminf_{x \to \infty} \frac{M(x)}{\sqrt{x} (\log \log \log x)^{5/4}} < 0 <\limsup_{x \to \infty} \frac{M(x)}{\sqrt{x} (\log \log \log x)^{5/4}} < \infty. $$

Using these results and conjectures to address the original question seems to be challenging, because of possible cancellation between $\frac{M(x)}{\sqrt{x}}$ and $\int_{1}^{x} \frac{M(t)}{t^{3/2}} \, dt$. My questions are the following:

Are known results about $M(x)$ enough to determine if $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

If not, does Gonek's conjecture (or any other plausible conjectures) imply that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?