Optimal universal conjugate-radius constant

Determine the optimal universal constant in the conjugate-radius bound for complete, connected, noncompact three-dimensional Riemannian manifolds without boundary whose scalar curvature satisfies \(\operatorname{Scal}_g\ge 6\), improving or confirming the bound \(\operatorname{conj}(M,g)\le 2\pi/3\).

Background

The paper proves that every complete, connected, noncompact Riemannian three-manifold without boundary with scalar curvature at least 6 has conjugate radius at most $2\pi/3$. After rescaling, the corresponding estimate is $\operatorname{conj}(M,g)\le 2\pi/(3\sqrt{\kappa})$ when $\operatorname{Scal}_g\ge 6\kappa>0$.

The authors note that the product manifold $S^2(1/\sqrt{3})\times\mathbb R$ has scalar curvature 6 and conjugate radius $\pi/\sqrt{3}$, providing a lower-bound example for any optimal universal constant. The exact best constant is left unresolved.