Voronoi's conjecture on parallelohedra

Let $P$ be a $d$-dimensional parallelohedron, meaning a convex polytope that tiles $\mathbb R^d$ by translations. A Voronoi polytope of a $d$-dimensional lattice $\Lambda$ is the set of points at least as close to the origin as to any other point of $\Lambda$. Voronoi's conjecture. For every $d$-dimensional parallelohedron $P$, there exists a $d$-dimensional lattice $\Lambda$ such that $P$ and the Voronoi polytope of $\Lambda$ are affinely equivalent. The conjecture is known in dimension five and for various classes of parallelohedra, but remains open in general.