Fractional part correlations and the Möbius function

We show that $$\sum _{n\mathrm{\ne }m}\frac{\mu (n)\mu (m)}{nm}{E}_X(\{nx\}\{mx\})=-\frac{9}{2{\pi }^2}+O\left(\frac{1}{X}\right),$$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$, $E_X(\mathrm{.})$ denotes the expected value, $\mu (\mathrm{.})$ denotes the Möbius function, and $\{\mathrm{.}\}$ denotes the fractional part function.