Robins' conjecture on Fourier transforms of convex centrally symmetric bodies

Let $\mathcal{P},\mathcal{Q}\subset \mathbb{R}^d$ be convex, centrally symmetric bodies. Suppose that $\mathcal{P}$ and $\mathcal{Q}$ are not multi-tilers, meaning that neither set $k$-tiles $\mathbb{R}^d$ with a translation set for any positive integer $k$. Their Fourier transforms are defined by

$$\widehat{1}_{\mathcal{P}}(\xi)=\int_{\mathbb{R}^d}1_{\mathcal{P}}(x)e^{-2\pi i\langle x,\xi\rangle}\,dx,$$

and similarly for $\mathcal{Q}$, where $1_{\mathcal{P}}$ and $1_{\mathcal{Q}}$ are the characteristic functions.

Robins' conjecture. If

$$\widehat{1}_{\mathcal{P}}(\xi)=\widehat{1}_{\mathcal{Q}}(\xi)$$

for every $\xi\in\mathbb{Z}^d$, then $\mathcal{P}$ and $\mathcal{Q}$ coincide up to rigid motions, except on a set of measure zero; equivalently, the two conditions hold if and only if the bodies coincide up to such motions.

The conjecture concerns whether lattice samples of the Fourier transform determine a non-multi-tiling convex centrally symmetric body up to rigid motion. The paper states that it constructs a counterexample, so the conjecture is refuted in the dimension covered by that construction; the supplied excerpt does not specify the dimension or the precise counterexample.