Periodicity conjecture for torsion-free crystallographic groups

Let $\Gamma<\operatorname{Isom}(\mathbb{R}^k)$ be a torsion-free crystallographic group. Its quotient $\mathbb{R}^k/\Gamma$ is a closed aspherical manifold, and $K^*(\mathbb{R}^k/\Gamma)$ denotes its complex topological $K$-theory. Periodicity conjecture. For $*>\mathbb{Q}\operatorname{cd}(\Gamma)-2$, there is an isomorphism

$$\pi_*K^{\mathrm{def}}(\Gamma)\cong K^*(\mathbb{R}^k/\Gamma).$$

For products of aspherical surface groups, analogous identifications are known above the rational cohomological dimension minus two. The conjecture predicts the corresponding description for torsion-free crystallographic groups, extending the known Bott periodicity result for virtually free abelian groups.