Nishikawa's classification conjecture for curvature operator of the second kind

Let $(M^n,g)$ be a closed Riemannian manifold. The curvature operator of the second kind is the curvature action restricted to traceless symmetric two-tensors $S^2_0(T_pM)$.

Nishikawa's conjecture.

1. If $M$ has positive curvature operator of the second kind, then $M$ is diffeomorphic to a spherical space form.
2. If $M$ has nonnegative curvature operator of the second kind, then $M$ is diffeomorphic to a Riemannian locally symmetric space.

The positive case has been settled: Cao, Gursky and Tran proved the stronger result under two-positive curvature operator of the second kind, using the strictly PIC1 condition and Ricci flow. The nonnegative case remains unresolved in the supplied text.