Littlewood's conjecture in k dimensions

Let $\mathbf{x}=(x_1,\ldots,x_k)\in\mathbb{R}^k$, where $k\geqslant2$. Littlewood's conjecture in $k$ dimensions.

$$\liminf_{n\to\infty}n\|nx_1\|\cdots\|nx_k\|=0.$$

This is the multiplicative form of simultaneous Diophantine approximation. Gallagher's theorem establishes the corresponding approximation rate for almost every vector, while the assertion for every vector remains open for $k\geqslant2$.