The universal spectrum conjecture for lattice tilings

Let $\Gamma=\mathbb{Z}^n+\mathcal A$, where $\mathcal A\subset\frac{1}{N_1}\mathbb{Z}\times\cdots\times\frac{1}{N_n}\mathbb{Z}$. Write $A=\mathcal A\pmod{\mathbb{Z}^n}$, and suppose that $A$ admits a factorization

$$A\oplus B=\mathbb{Z}_{N_1}\times\cdots\times\mathbb{Z}_{N_n}.$$

A set $\Lambda$ is a universal spectrum for $\Gamma$ if it is a spectrum simultaneously for every bounded measurable set tiled by $\Gamma$.

Universal spectrum conjecture. Under these assumptions, $\Gamma$ has a universal spectrum of the form

$$\Lambda=N_1\mathbb{Z}\times\cdots\times N_n\mathbb{Z}+\mathcal L,$$

with $\mathcal L\subset\mathbb{Z}^n$.

This conjecture concerns the existence of one spectrum working uniformly for all bounded measurable sets with a prescribed lattice-type tiling set. The supplied text gives no resolution status.