The strong regularity-radius conjecture for Delone sets

Let $X$ be a Delone set in $\mathbb{R}^d$ with parameters $(r,R)$, where $R$ is the radius of a largest empty ball, and let $\hat{\rho}_d=\hat{\rho}_d(r,R)$ be the smallest radius such that every Delone set with congruent clusters of that radius is a regular system. Strong Conjecture. There exists a constant $c$, independent of $d$ and $R$, such that

$$\hat{\rho}_d\leq c(d\log d)R$$

for each $d\geq 1$; equivalently, the bound is independent of $r$. This is a stronger proposed asymptotic bound than the weak conjecture and is motivated by known linear lower bounds and plausibility considerations, but it remains open.