The linear repetitivity conjecture for dense repetitivity

Let $X$ be a Delone set. Its patch-counting function $N_X(T)$ counts translation-equivalence classes of $T$-patches, and its repetitivity function $M_X(T)$ is the least radius in which every $T$-patch occurs. The set is linearly repetitive if $M_X(T)=O(T)$ as $T\to\infty$, and densely repetitive if $M_X(T)=O((N_X(T))^{1/n})$ as $T\to\infty$.

The linear repetitivity conjecture. Every aperiodic linearly repetitive Delone set is densely repetitive.

Equivalently, every aperiodic linearly repetitive set $X$ in $\mathbb R^n$ should satisfy $\liminf_{T\to\infty}N_X(T)/T^n>0$. The conjecture was proved by Lenz after the paper was written.