Periodic approximability of cut-and-project Delone subshifts

Let $\mathcal{D}\in Del_{U,K}^0$ be a Delone set whose transversal is considered in the associated subshift. A Delone set is defined via a cut-and-project scheme when it arises by cutting a strip in a higher-dimensional space and projecting the relevant lattice points to the physical space.

Periodic approximability conjecture. The transversal of $\mathcal{D}$ is periodically approximable if $\mathcal{D}$ is defined via a cut-and-project scheme.

This conjecture concerns periodic approximations of model-set subshifts, a class of quasicrystals with connections to substitutional systems. The source presents it as a proposed direction for future work; no resolution is given.