Separated-subgroup conjecture for discrete torus actions

Let $n\in\mathbb{N}$ and $D>0$. A pointed simply-connected proper geodesic space is a pair $(X,p)$, and $\operatorname{Iso}(X)$ denotes its isometry group. For a discrete subgroup $G\leq\operatorname{Iso}(X)$, the quotient $X/G$ has its quotient metric. Separated-subgroup conjecture. For every $n\in\mathbb{N}$ and $D>0$, there exists $D'(n,D)>0$ such that, whenever $(X,p)$ is a pointed simply-connected proper geodesic space and $G\leq\operatorname{Iso}(X)$ is discrete with

$$G\cong\mathbb{Z}^n,\qquad \operatorname{diam}(X/G)\leq D,$$

there is a finite-index subgroup $\Gamma\leq G$ satisfying

$$\operatorname{diam}(X/\Gamma)\leq D'$$

and

$$d(gp,p)\geq D\quad\text{for all }g\in\Gamma\setminus\{e\}.$$

The conjecture asks whether a uniformly separated finite-index sublattice can always be found while retaining a uniformly bounded quotient diameter; it is presented in the paper as an interesting problem related to removing a pre-compactness hypothesis, and no resolution is supplied there.