Are there any algebraic integers of degree $d \geq 3$ with bounded partial quotients?

It is a theorem of Dirichlet that for every irrational number $\alpha$, there exists infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1$ and $q > 0$ such that

$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^2}.$

This can be improved with the constant on the right hand side improved to $1/\sqrt{5}$, and then this is sharp (Hurwitz's theorem). The reason is that there are badly approximable numbers, whose partial quotients in their continued fraction expansions are bounded, for which it is possible to prove a lower bound of the form $|\alpha - p/q| \geq c(\alpha) q^{-2}$ for all rational numbers $p/q$ and the constant $c(\alpha)$ depending only on $\alpha$. Note that since quadratic irrationals have eventually periodic continued fraction expansion, all quadratic irrationals are badly approximable.

It is a theorem of Roth, for which he was awarded a Fields Medal in 1958, that for any algebraic integer $\alpha$ having degree $d \geq 2$ and for any $\varepsilon > 0$, the number of reduced fractions $p/q$ such that

$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \varepsilon}}$

is finite. In other words, all algebraic integers are almost badly approximable.

The question is, are there any algebraic integers $\alpha$ having degree $d \geq 3$ which has bounded partial quotients, or equivalently, badly approximable? This question, shockingly, remains open even for degree 3.