Nishikawa's curvature operator of the second kind conjecture

Let $(M,g)$ be a closed, simply connected Riemannian manifold. Let $d40$ denote the curvature operator of the second kind, restricted as a bilinear form to the trace-free symmetric two-tensors $S^2_0(TM)$. Nishikawa's conjecture. If

$$\hat{\mathrm R}\geq 0,$$

then $M$ is diffeomorphic to a Riemannian locally symmetric space. If the inequality is strict, then $M$ is diffeomorphic to a round sphere. The paper's main result settles the strict-positive sphere assertion by showing that manifolds with positive curvature operator of the second kind satisfy Brendle's PIC1 condition; the nonnegative case is not resolved here.