Nonexistence conjecture for LR cubical and canonical cut-and-project sets with convex shapes

Let $d>1$, and let $\mathcal{C}_d'$ be the collection of convex sets in $\mathbb{R}^d$ with inradii at least $1/2$. A cut-and-project set is called cubical or canonical as in the paper, and $\mathrm{LR}$ and $\mathrm{LR}_\Omega$ denote the corresponding repetitivity properties with respect to a collection of shapes. Nonexistence conjecture. For $d>1$, there are no cubical or canonical cut-and-project sets which are $\mathrm{LR}$ or $\mathrm{LR}_\Omega$ with respect to $\mathcal{C}_d'$. The question is presented as an open problem for future work; the authors explain that their preceding arguments do not settle it.