Almost-divisibility conjecture for integers in a power interval

Let $0<\alpha\leq1/2$ and let $X$ be a sufficiently large integer. Given constants $0<c_1<c_2\leq1$, consider the least $L$ such that $[X-L,X]$ contains an integer divisible by some integer in $[c_1X^\alpha,c_2X^\alpha]$.

Almost-divisibility conjecture. For every $\epsilon>0$, one can take $L=X^\epsilon$ provided $X$ is sufficiently large in terms of $\epsilon$.

This asks whether every sufficiently large endpoint can be approximated within an arbitrarily small power error by an integer having a divisor in the specified interval. The supplied text presents this as a suspicion and gives no resolution.