The directional $p$-adic Littlewood conjecture for vectors

Let $\overline{\alpha}\in\mathbb{R}^n$ be a vector and let $p$ be a prime. Denote by $\langle\cdot\rangle:\mathbb{R}^n\to[0,1)^n$ the $\ell^\infty$ distance to the nearest integer vector, and write $\|\cdot\|_\infty$ for the $\ell^\infty$ norm. The directional $p$-adic Littlewood conjecture.

$$\liminf_{k\to\infty}(k|k|_p)^{1/n}\left\|\langle k\overline{\alpha}\rangle\right\|_\infty=0.$$

This is proposed as a natural vector-valued extension of the one-dimensional $p$-adic Littlewood conjecture. The source says that it lacks a reference in the literature and provides no resolution, so it remains open.