Short-interval variance conjecture for sums of the Möbius function times the number-of-prime-factors function

Let $\mu(n)$ denote the Möbius function and let $\omega(n)$ denote the number of distinct prime factors of $n$. For $X\to\infty$, let $H=X^\delta$ with $\delta\in(0,1)$. Short-interval variance conjecture for $\mu\omega$.

$$\frac{1}{X}\int_X^{2X}\Big(\sum_{x\leq n\leq n+H}\mu(n)\omega(n)\Big)^2\,dx\sim H(\log\log X)^2.$$

The source presents this as an integer analogue of a proved function-field variance formula. It is motivated by the growth $H_n\sim\log n$ with $n=\deg(f)$, but its status is not specified.