Cassels–Swinnerton-Dyer's conjecture on periodic multidimensional continued fractions

Let $d\geqslant3$ and let $\Lambda$ be an irrational full-rank lattice in $\mathbb{R}^d$. Let $\mathcal{K}$ be one of the $2^n$ Klein polyhedra of $\Lambda$. The facets and edge stars of $\mathcal{K}$ have determinants as defined in the source, and $(d-1)$-periodicity refers to periodicity of the boundary's combinatorial structure together with these determinants.

Cassels–Swinnerton-Dyer's conjecture. The following are equivalent:

1. The facets and the edge stars of the vertices of $\mathcal{K}$ have bounded determinants, bounded by a common constant.
2. The combinatorial structure of the boundary of $\mathcal{K}$, equipped with the determinants of facets and edge stars of vertices, is $(d-1)$-periodic.

This is presented as a reformulation of the positive-norm-minimum conjecture: bounded multidimensional analogues of partial quotients should imply periodicity of the associated multidimensional continued fraction. The supplied status is unknown, so the database records it as open.