Generalized Einstein rigidity conjecture for pointwise smaller metrics

Let $(M^n,g_E)$ be a closed Einstein $n$-manifold with $n\geq 3$ and negative scalar curvature $R^*<0$. Let $g$ be another metric on $M$ satisfying

$$\operatorname{Sc}_g\geq R^*,\qquad g\leq g_E.$$

Here $g\leq g_E$ means that $g(v,v)\leq g_E(v,v)$ for every tangent vector $v$. Generalized Einstein rigidity conjecture. The metric $g$ is isometric to the Einstein metric $g_E$.

This removes the harmonicity and auxiliary condition from the preceding rigidity theorem and extends the proposed rigidity principle beyond the hyperbolic case. The source provides no resolution of the conjecture.