Roth's theorem states that for every real algebraic $\alpha$ and $\epsilon>0$, there is a $c>0$ such that $$|\alpha -\frac{p}{q}| > \frac{c}{q^{2+\epsilon}}.$$

Lang's conjecture strengthened this to $$|\alpha -\frac{p}{q}| > \frac{c}{q^2 (\log q)^{1+\epsilon}}.$$

A naive further strengthening would be to ask for $$|\alpha -\frac{p}{q}| > \frac{c}{q^2},$$ and this is satisfied by all $\alpha$ with bounded partial quotients $a_n$ (Khinchin Thm 23).

Life would be nice and simple if it were true that all algebraic $\alpha$ has bounded $a_n$. This seems too much to be asking for but I am ignorant of any counterexample. Does anyone knows a counter example or even some heuristic reasoning why this is unlikely to be true ? Thanks in advance.