The patch-counting lower-bound conjecture for aperiodic Delone sets

Let $X$ be a Delone set in $\mathbb R^n$, with Delone constants $(r,R)$, meaning that distinct points of $X$ are separated by at least $r$ and every point of $\mathbb R^n$ lies within distance $R$ of $X$. The set is aperiodic if it has no nonzero translational period, and $N_X(T)$ denotes the number of translation-equivalence classes of $T$-patches.

The patch-counting lower-bound conjecture. For every dimension $n\geq 1$ and Delone constants $(r,R)$, there is a positive constant $c=c(n,r,R)$ such that every aperiodic Delone set $X$ in $\mathbb R^n$ with constants $(r,R)$ satisfies

$$\limsup_{T\to\infty}\frac{N_X(T)}{T^n}\geq c(n,r,R).$$

This conjecture gives a uniform lower bound on patch complexity for aperiodic Delone sets with fixed Delone constants. It is presented as strengthening the corresponding conjecture where the constant may depend on $X$; the paper states no resolution.