Gromov's noncompact injectivity radius conjecture

Let $(M^n,g)$ be a complete, non-compact Riemannian manifold with uniformly positive scalar curvature

$$\operatorname{Sc}(g)\geq (n-1)(n-2).$$

Gromov's noncompact injectivity radius conjecture. Then

$$\operatorname{Inj}(M)\leq \pi,$$

and equality holds if and only if $M$ is isometric to

$$(\mathbb S^{n-1}(1)\times\mathbb R, g_{\mathbb S^{n-1}}+dt^2).$$

The source notes that even the weaker bound $\operatorname{Inj}(M)\leq\pi$ is open for complete noncompact manifolds with uniformly positive scalar curvature. The conjecture is presented as a stronger rigidity statement.