The 5-gonal symmetry conjecture for Delone sets

Let $X\subset\mathbb R^3$ be a Delone set with parameter $R$. For each point $x\in X$, let $S_x(2R)$ be its local group, and let $Y$ be the subset of points whose local groups contain no 5-fold axes. 5-gonal symmetry conjecture. Given a Delone set $X\subset\mathbb R^3$, the subset $Y$ of points $x$ whose groups $S_x(2R)$ are free of 5-fold axes is also a Delone set. This conjecture is presented as a consequence of the crystalline kernel conjecture, since $K\subseteq Y$; the source gives no resolution of either claim.