Oppenheim's conjecture on lattices with positive norm minimum

Let $n\geqslant 3$ and let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice with positive norm minimum $N(\Lambda)>0$. A lattice is called algebraic when it is similar, modulo the action of the group of diagonal $n\times n$ matrices, to the lattice of a complete module of a totally real algebraic field of degree $n$. Oppenheim's conjecture. The lattice $\Lambda$ is algebraic.

The converse is an obvious consequence of the Dirichlet theorem on algebraic units, while the conjecture itself remains unproved. In dimension three it implies Littlewood's conjecture, and attempted proofs in dimensions three and higher contained an essential gap.