The conjecture for the variance of squarefull numbers in short intervals

Let $Q(t)$ denote the counting function of squarefull numbers up to $t$. Given $\epsilon>0$, $X>1$, and $X^\epsilon\leq H\leq X^{1/4-\epsilon}$, set

$$y=2\sqrt{x}H+H^2.$$

Variance conjecture.

$$\frac{1}{X}\int_X^{2X}\left|Q(x+y)-Q(x)-\frac{\zeta(3/2)}{\zeta(3)}H\right|^2\,dx\sim \frac{4\zeta(4/3)}{3\zeta(2)}\int_0^\infty\left(\frac{\sin \pi y}{\pi y}\right)^2y^{1/3}\,dy\cdot H^{2/3}.$$

This predicts the asymptotic size and leading constant of the mean-square fluctuation in the count of squarefull numbers over intervals whose endpoint difference satisfies $\sqrt{x+y}-\sqrt{x}=H$. The preceding result gives an upper bound in a shorter-range variance problem, while this sharper asymptotic remains conjectural in the stated range.