Conjecture on reciprocal fractional-part sums for algebraic irrationals

For an irrational real number $\alpha$, define

$$S_N(\alpha,0)=\sum_{1\le n\le N}\frac{1}{n\|n\alpha\|}.$$

Algebraic-irrational conjecture. For every algebraic $\alpha\in\mathbb{R}\setminus\mathbb{Q}$,

$$S_N(\alpha,0)\asymp(\log N)^2.$$

The estimate would show that algebraic irrationals have the same order of growth as almost every irrational in this problem. The paper explains that establishing it is equivalent to a quadratic average-growth bound for the associated continued-fraction quantities, but gives no proof.