The variance asymptotic conjecture for squarefull numbers in short intervals

Let $Q(t)$ denote the number of squarefull numbers at most $t$. Given $\epsilon > 0$, $X > 1$ and $H < X^{1/2 - \epsilon}$, set

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

Variance asymptotic conjecture. The variance of the count of squarefull numbers in these intervals satisfies

$$\frac{1}{X} \int_{X}^{2 X} \Big|Q(x + y) - Q(x) - \frac{\zeta(3/2)}{\zeta(3)} H \Big|^2 dx \sim \frac{\zeta(4/3)}{\zeta(2)} \int_{0}^{\infty} \Bigl(\frac{\sin \pi y}{\pi y}\Bigr)^2 y^{1/3} dy \cdot H^{2/3}.$$

This conjecture predicts the true order and leading constant for the variance, improving substantially on the paper's upper bound. The stated range is that of the conjecture; the source gives no resolution, so it remains open.