Algebraic continued-fraction partial-quotient conjecture

Let $\alpha=[a_0;a_1,a_2,\dots]\in\mathbb{R}\setminus\mathbb{Q}$ be an algebraic irrational, with continued-fraction partial quotients $a_k$.

Algebraic partial-quotient conjecture. For every such $\alpha$,

$$\sum_{k=1}^n a_k\ll n^2.$$

The source states that this is equivalent to a conjectured estimate for certain Diophantine sums and notes computational evidence for specific algebraic numbers. It remains open for general algebraic irrationals.