Let $0 < \theta < 1$ be a fixed real number, and $0 < c_1 < c_2 < 1$ be real numbers. For a positive integer $n$, let $d_{\theta, c_1, c_2}(n)$ be the number of divisors $k$ of $n$ satisfying

$$\displaystyle c_1 n^\theta < k < c_2 n^\theta.$$

My question is, how large can $d_{\theta, c_1, c_2}(n)$ as a function of $n$? In particular, can $d_{\theta, c_1, c_2}(n)$ be larger than $(\log n)^A$ for any positive number $A$?