The p-adic Littlewood conjecture

Let $p$ be a prime and let $b1$ be an irrational real number. Let $\lvert q\rvert_p=p^{-v_p(q)}$ denote the $p$-adic absolute value of the integer $q$, where $v_p(q)=\max\{k:p^k\mid q\}$. The $p$-adic Littlewood conjecture. For every prime $p$ and irrational real $\alpha$, one has

$$\inf_{q\geq 1}q\cdot\lvert q\rvert_p\cdot\lVert q\alpha\rVert=0.$$

This is a special case of the mixed Littlewood conjecture and is equivalent, for $p=2$, to the $2$-adic Littlewood problem discussed in the paper. Its status is not resolved in the supplied text.