Littlewood's conjecture for products of linear forms

Let $f$ be a product of $k$ linear forms in $\mathbb R^k$, where $k \geqslant 3$, and assume that $f$ is not proportional to a multiple of a polynomial with integer coefficients. Littlewood's conjecture for products of linear forms.

$$\inf \{ |f(\mathbf x)|: {\boldsymbol 0} \ne \mathbf x \in \mathbb Z^k \} = 0.$$

Cassels and Swinnerton-Dyer derived Littlewood's conjecture from the case $k=3$. The stated conjecture is a special case of Margulis's conjecture and remains open in general.