Given an irrational algebraic number $\alpha$ (and maybe I want to add: of degree greater than $2$?), do there exist infinitely many relatively prime and *square-free* $p$,$q$ with
$$|\alpha - p/q | < \frac{1}{q^2}\ .$$

I would guess "yes" based on a combination of Dirichlet's approximation theorem and the positive density of square-free integers among all the integers, but I can't even think of one example where I can prove this.