The weak 5-gonal symmetry conjecture for Delone sets

Let $X\subset\mathbb R^3$ be a Delone set with mutually equivalent $2R$-clusters. For each point $x\in X$, let $S_x(2R)$ denote its local group. Weak 5-gonal symmetry conjecture. The local group $S_x(2R)$ contains no 5-fold axis. This is stated as an easier consequence enforced by the 5-gonal symmetry conjecture in the case of mutually equivalent $2R$-clusters; the source gives no resolution of the claim.