Generic non-embeddability conjecture for cut-and-project quasicrystal tiling spaces

Let $E_\parallel$ be a $d$-dimensional linear subspace of ${\mathbb R}^n$ with irrational orientation, and let $\mathcal L(E_\parallel)$ be the associated cut-and-project Delone set. Equip its tiling space with the combinatorial metric, where the distance between two tilings is the inverse of the largest radius on which their patches at the origin coincide. Generic non-embeddability conjecture. For Lebesgue almost all $d$-dimensional linear spaces $E_\parallel\in{\mathbb R}^d$, viewed as points in the Grassmannian manifold of $d$-dimensional subspaces of ${\mathbb R}^n$, the tiling space of $\mathcal L(E_\parallel)$, endowed with the combinatorial metric, is not $f$-embeddable. This is proposed by analogy with the Sturmian result, where $f$-embeddability is characterized by bounded type; the source gives no resolution of the generic higher-dimensional claim.