A conjugate radius bound for open three-manifolds

We prove that every complete, connected, noncompact Riemannian three-manifold without boundary with scalar curvature at least $6$ has conjugate radius at most $2\pi \mathrm{/}3$. This resolves Shen’s conjecture on the strict conjugate-radius bound $\mathrm{conj}(M,g)<\pi$ in dimension three.