Untwisted algebraic 1-torsion for higher-dimensional contact manifolds without Giroux torsion

Let $Y$ be the unit cotangent bundle of a hyperbolic surface, and let $(M=\Sigma \times Y,\xi_k)$ be the corresponding $5$-dimensional contact manifold from Corollary

$$. **Untwisted algebraic 1-torsion conjecture.** The examples of Corollary$$

have untwisted algebraic $1$-torsion for any $k\geq 1$. For $k\geq 3$, this would give an infinite family of contact manifolds with no Giroux torsion and algebraic $1$-torsion in dimension $5$. The conjecture is motivated by geometric considerations and by the relationship between SFT and string topology; its resolution is not given here.