Let $\mu$ denote the Möbius function, and let $\chi$ be a primitive Dirichlet character of modulus $q$. Define $M(n, \chi)=\sum_{j \leq n} \mu(j)\chi(j)$ and

$$f(x, \chi):=\sum_{n \leq x} \frac{(-1)^{n-1}\overline{\chi}(n)M(n, \chi)}{n}.$$

Fix $\varepsilon >0$. Is $f(x, \chi)=\Omega(x^{\frac{1}{2}-\varepsilon})$ for infinitely many $x \rightarrow \infty$ ?

According to the answer to my previous question: https://mathoverflow.net/a/444921/501735, this seems to be the case, since $\chi(n)=0$ only if $(n, q)=1$ and there seems to be at least $x^{1-2\varepsilon}$ even positive integers $n \leq x$ such that $M(n, \chi) < - n^{\frac{1}{2}-\varepsilon}$.