Variance of squarefull numbers in short intervals II

In this paper, we continue the study on variance of the number of squarefull numbers in short intervals $(x, x + 2 \sqrt{x} H + H^2]$ with $X \le x \le 2X$. We obtain the expected asymptotic for this variance over the range $X^\epsilon \le H \le X^{0.180688...}$ unconditionally and over the optimal range $X^\epsilon \le H \le X^{0.25 - \epsilon}$ conditionally on the Riemann Hypothesis or the Lindel\"{o}f Hypothesis.