Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?

For example, are there hyperbolic 5-manifolds that have the same integer cohomology ring as $\mathbf{R}P^5$, or another 5d lens space?