Dual form of Littlewood's conjecture in d dimensions

Let $H(\mathbf m)$ denote the multiplicative height of $\mathbf m \in \mathbb Z^d$, and let ${\boldsymbol{\alpha}} \in \mathbb R^d$ with $d \geqslant 2$. Dual form of Littlewood's conjecture.

$$\inf_{{\boldsymbol 0} \ne \mathbf m \in \mathbb Z^d} H(\mathbf m) \| \mathbf m \cdot {\boldsymbol{\alpha}} \| = 0.$$

The paper states that this formulation is equivalent to Littlewood's conjecture, citing Cassels–Swinnerton-Dyer and Dani–Margulis-type results. It remains open for $d \geqslant 2$.