Higher-dimensional linear repetitivity conjecture for tiling spaces

Let $A$ be a finite alphabet, and let a tiling in ${\mathbb R}^d$ have a suitably defined notion of linear repetitivity. Its transversal is the tiling space consisting of tilings locally equivalent to it and marked at the origin. Higher-dimensional linear repetitivity conjecture. There is a suitable definition of linearly repetitive tiling in ${\mathbb R}^d$ such that the transversal of any such tiling space is $f$-embeddable. This proposes an extension of the one-dimensional theorem that the tiling space of a linearly repetitive sequence is $f$-embeddable; the source does not provide a resolution of the higher-dimensional claim.