On divisor bounded multiplicative functions in short intervals

In 2016, the celebrated Matomaki-Radziwill theorem showed that there are cancellations for 1-bounded multiplicative functions in almost all short intervals. Our recent work proved that Matomaki-Radziwill theorem can be extended to divisor bounded multiplicative functions. Especially, we proved that for any sufficiently large $X$, $\epsilon>0$ and $h \geq (\log X)^{(1+\epsilon)k \log k- k +1}$, 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 $x \in [X,2X]$, where $d_k(n) = \sum_{m_1 \cdots m_k = n} 1$.