Classification of conservation-forced primitive substitution tilings

Let $(\mathcal{T},\sigma,M)$ be a primitive substitution tiling system of $\mathbb{R}^d$ with finite local complexity. Call it conservation-forced when it satisfies the conservation-law conditions defined earlier in the paper. Let $\lambda_{\mathrm{PF}}(M)$ denote the Perron–Frobenius eigenvalue of the substitution matrix $M$. The tiling is also assumed, in the second condition below, to arise from a lattice $\Lambda\subset\mathbb{R}^N$ as a canonical projection tiling satisfying the paper's Definition~. Classification conjecture. The following conditions are equivalent:

1. $(\mathcal{T},\sigma,M)$ is conservation-forced.
2. $\mathcal{T}$ is a primitive-substitution canonical projection tiling arising from a lattice $\Lambda\subset\mathbb{R}^N$ and satisfying Definition~, and $\lambda_{\mathrm{PF}}(M)$ is a Pisot number.

The implication from condition 2 to condition 1 is proved, even without the Pisot assumption, whereas the converse implication remains open.