On the smooth rigidity of almost-Einstein manifolds with nonnegative isotropic curvature

Let $(M^n,g)$, $n\ge 4$, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\text{eps}=\text{eps}(l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies $$l\le s\le L$$ and the Einstein tensor satisfies $$\mathrm{\mid }Ric-\frac{s}{n}g\mathrm{\mid }\le \text{eps}$$ then $M$ is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result of S. Brendle that a compact Einstein manifold with nonnegative isotropic curvature is isometric to a locally symmetric space.