Divergence of $\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right)$, where $\gamma$ is the Euler-Mascheroni constant

Question

We denote the Möbius function as $\mu(n)$, see its definition from this MathWorld.

On the other hand it is not known if the Euler-Mascheroni constant is irrational.

After I've read a MathOverflow post I would like to ask a related question.

Question. Is feasible to deduce if $$\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right),\tag{1}$$ is divergent? I am asking about what work can be done. Many thanks.

If you want to add a some words or a draft about if our series can be summable by any Cesàro means it also is good.