Unique ergodicity and uniform density for compact-window model sets

Let $(G\times H,\mathcal{L})$ be a cut and project scheme, let $W\subset H$ be a compact set, and let \text{\Large \curlywedge}(W) denote the associated model set. Write $\theta_H$ for Haar measure on $H$, and let $\underline{\operatorname{dens}}(\varLambda)$ and $\overline{\operatorname{dens}}(\varLambda)$ denote the lower and upper uniform densities of the model set \varLambda=\text{\Large \curlywedge}(W). Unique ergodicity conjecture. If \text{\Large \curlywedge}(W) is uniquely ergodic, then

$$\theta_H(W^\circ)=\underline{\operatorname{dens}}(\varLambda)\quad\text{and}\quad\overline{\operatorname{dens}}(\varLambda)=\theta_H(W).$$

The proposed equalities would characterize the boundary condition $\theta_H(\partial W)=0$, which is central to characterizing regular model sets, through uniform density. The statement is motivated by the preceding sharpness example and is presented as a reasonable expectation; no resolution is supplied in the source.