Continuum many bounded displacement non-equivalences for substitution tiling spaces

Let $\varrho$ be a primitive substitution rule in $\mathbb{R}^d$, with eigenvalues ordered as in Theorem 1, and let $t\geq 2$ be the minimal index for which $\lambda_t$ has an eigenvector whose sum of coordinates is non-zero. Let $\mathbb{X}_\varrho$ denote the associated substitution tiling space, and let $\operatorname{BD}(\mathbb{X}_\varrho)$ denote its set of bounded-displacement equivalence classes.

Continuum-many BD classes conjecture. If

$$|\lambda_t|>\lambda_1^{\frac{d-1}{d}},$$

then

$$|\operatorname{BD}(\mathbb{X}_\varrho)|=2^{\aleph_0}.$$

The theorem preceding this conjecture proves the same conclusion under an additional hypothesis involving two legal patches whose supports differ by a translation and whose tile-count difference is not orthogonal to a suitable eigenvector. The conjecture predicts that this patch hypothesis is unnecessary; the bounded-displacement criterion suggests that the eigenvalue-growth condition alone forces continuously many pairwise non-equivalent tilings.