The sharp symmetry-rank conjecture for cyclic fundamental groups

Let $M^{2m-1}$ be a closed Riemannian manifold with positive sectional curvature. Let $q$ denote the smallest prime divisor of $m$, and let an $r$-torus act effectively and isometrically on $M$. Sharp cyclicity conjecture. If

$$r \geq \frac{m}{q}+1,$$

then $\pi_1(M)$ is cyclic. This conjecture would give a sharp obstruction to non-cyclic fundamental groups under a sufficiently large torus symmetry rank; the bound is motivated by spherical space-form examples, which show that the corresponding threshold cannot generally be lowered.