The adjacency-component conjecture for perfect Delaunay polytopes

Let $P$ and $P_1$ be perfect Delaunay polytopes for $\mathbb{Z}^n$ and call them adjacent when the quadratic rank of their common vertex set is $2$. An adjacency component is a collection of perfect Delaunay $n$-polytopes connected by a sequence in which every two consecutive members are adjacent. Adjacency-component conjecture. For any $n e\in\mathbb{N}$, all perfect Delaunay $n$-polytopes belong to the same adjacency component. The conjecture is motivated by the authors' finding that, for each $n\leq 8$, all known perfect Delaunay $n$-polytopes lie in one adjacency component; its general status is not resolved in the provided text.