On the asymptotics of some sum involving the Mertens function

Question

Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \mu(n)$. My question is, how large can $S(x)=\sum_{\sqrt x < n < x} a_{n}M(x/n)$ be ? I would ``conjecture'' that $|S(x)|= \Omega(x^{\Theta-\epsilon})$ for infinitely many $x\rightarrow \infty$, $\Theta$ being the supremum of the real parts of the zeros of the Riemann zeta function.