On divisor bounded multiplicative functions in short intervals

Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in \mathbb{N}$. In this paper we improve Mangerel's results which extend the Matom\"aki-Radziwi{\l\l} theorem to divisor bounded multiplicative functions. In particular, we prove that for sufficiently large $X \geq 2$, any $\epsilon>0$ and $h \geq (\log X)^{k \log k - k + 1 + \epsilon}$ , we have $$ \frac{1}{h}\sum_{x<n\leq x+h}d_k(n)-\frac{1}{x}\sum_{x<n\leq 2x}d_{k}(n) = o(\log^{k-1} x) $$ for almost all $x \in [X,2X]$. We also demonstrate that the exponent $k \log k-k+1$ is optimal.
Comment: The author thanks Alexander Mangerel and Aled Walker for their useful comments