The 2-adic Littlewood conjecture in continued-fraction form

For an irrational real number $\alpha=[a_0;a_1,a_2,\ldots]$, let $M(\alpha)=\sup_{n\geq 1}a_n(\alpha)$, where $a_n(\alpha)$ are its continued-fraction partial quotients. The 2-adic Littlewood conjecture in continued-fraction form. For every irrational real $\alpha$, one has

$$\sup_{k\geq 0}M(2^k\alpha)=\infty.$$

The paper explains that this is equivalent to the $p$-adic Littlewood conjecture for $p=2$: a counterexample to the latter exists exactly when the displayed supremum is finite. The supplied text gives no resolution of this formulation.