The higher-dimensional torus contact-structure fundamental group conjecture

Let $T^{2n+1}$ be the $(2n+1)$-dimensional torus, let $\Xi(T^{2n+1})$ denote its space of contact structures, and let $\xi$ be a contact structure on $T^{2n+1}$. A contact structure is $T^2$-invariant when it is invariant under the relevant $T^2$-action on $T^{2n+1}$.

Higher-dimensional torus conjecture. There exists a $T^2$-invariant contact structure $\xi$ on $T^{2n+1}$ such that

$$\pi_1(\Xi(T^{2n+1}),\xi)$$

based at $\xi$ contains a subgroup isomorphic to $\mathbb{Z}^{2n-1}$.

This extends the preceding construction and computation for $T^5$ to higher-dimensional odd-dimensional tori. Establishing the conjecture requires controlling contractible closed Reeb orbits for a $T^2$-invariant contact form so that cylindrical contact homology is well-defined.