The B-variant of the 2-adic Littlewood conjecture

For an irrational real number $\alpha=[a_0;a_1,a_2,\ldots]$, let $B(\alpha)=\limsup_{n\to\infty}a_n(\alpha)$, where $a_n(\alpha)$ are its continued-fraction partial quotients. The B-variant of the 2-adic Littlewood conjecture. For every irrational real $\alpha$, one has

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

Since $B(\alpha)\leq M(\alpha)$, the paper notes that this would be stronger than the original $2$-adic Littlewood conjecture. The authors state that they are not confident this claim is true, but formulate it as a conjecture; no resolution is supplied.