Fix constants $1\leq \alpha<\beta$. What is the density of the set of positive integers $n$ with at least two factors between $\alpha\sqrt{n}$ and $\beta\sqrt{n}$?

(I am specifically interested when $\alpha=\sqrt{2}$ and $\beta=\sqrt{3}$, and I am hoping the density is zero. I am not an expert in this field, so apologies in advance.)