The short-interval divisor conjecture for arbitrary fixed scales

Let $n\geq 1$ be an integer, let $\mathcal{D}_n$ be the set of positive divisors of $n$, and define

$$D_n(X,Y)=\left|\{d\in\mathcal{D}_n:X\leq d\leq X+Y\}\right|.$$

Fix real numbers $\theta$ and $\epsilon$ satisfying $0<\epsilon<\theta<1$. Short-interval divisor conjecture. There exists a constant $k_\epsilon(\theta)$ such that, for every integer $n\geq 1$,

$$D_n(n^\theta,n^{\theta-\epsilon})\leq k_\epsilon(\theta).$$

This proposes a uniform bound for divisors in intervals of length $n^{\theta-\epsilon}$ beginning at $n^\theta$, simultaneously for every fixed scale $0<\theta<1$. The supplied text does not state that the conjecture has been proved or disproved.