The rational-matrix Fuglede–Gabor conjecture

Let $K\subset \mathbb R^d$ and let $B$ be a rational matrix with $\det(B)=1$. Let $s$ be the least common multiple of the denominators of the entries $b_{ij}$. Rational-matrix Fuglede–Gabor conjecture. There is a matrix $C$ such that, for $\Lambda=\left(\begin{array}{cc}I&D\\ C&B\end{array}\right)(\mathbb Z^{2d})$, the system $\mathcal G(|K|^{-1/2}\chi_K,\Lambda)$ is a Gabor orthonormal basis for $L^2(\mathbb R^d)$ if and only if $K$ tiles and

$$K=\bigcup_{i=1}^sE_i=\bigcup_{i=1}^sF_i,$$

where each $E_i$ and $F_i$ is a fundamental domain of $\mathbb Z^d$ and $B^{-t}(\mathbb Z^d)$, respectively. This is posed in the rational-lattice case and is intended to characterize when a union of fundamental domains yields a Gabor orthonormal basis. The provided excerpt gives no resolution status.