Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Pick some positive $X,h \in \mathbb{R}$ and consider the sum

$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$

In view of Huxley's results, we can give the correct asymptotic estimate (as $X \to \infty$) for $S(X,h)$ once $h \gg X^{0.315}$. If I recall correctly, Shiu proved the correct upper bound for $S(x,h)$ even when $h \sim x^\epsilon$.

For small $h$, say $h \sim X^{0.2}$, can we obtain the correct asymptotic for $S(X,h)$?

It seems to me that the methods (Poisson summation, etc.) used to estimate the full sum should be applicable here, so I would like to know whether anyone has already carried this out carefully.

I am less interested in bounds that are valid for "most" values of $X$.