Cassels–Swinnerton-Dyer's conjecture on positive norm minima and algebraic lattices

Let $d\geqslant3$ and let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$. Its norm minimum is

$$N(\Lambda)=\inf_{\mathbf{z}\in\Lambda\setminus\{\mathbf{0}\}}\Pi(\mathbf{z})^d.$$

A lattice is called algebraic when it satisfies the algebraic-lattice condition referred to in the source. A diagonal operator is non-degenerate when its diagonal entries are nonzero.

Cassels–Swinnerton-Dyer's conjecture. The condition $N(\Lambda)>0$ is equivalent to the existence of a non-degenerate diagonal operator $D$ such that $D\Lambda$ is an algebraic lattice.

This is described as an analogue of Oppenheim's conjecture for decomposable forms and is stated to imply Littlewood's conjecture. The source gives a counterexample in dimension $2$ and leaves the assertion for $d\geqslant3$ as the conjectural case.