Closed counterexamples to Toponogov's question on mixed curvature

We construct explicit closed Riemannian manifolds in every even dimension carrying one-dimensional totally geodesic foliations with positive mixed sectional curvature. The leaves are closed geodesics and form smooth circle fibrations. In dimensions $4m+2$ the examples are constructed on $S^{4m-1}\times S^3$; in dimensions $4m$ they are obtained as totally geodesic fixed point submanifolds $S^{4m-1}\times S^1$. Since the normal rank is odd in all examples, the Ferus–Adams bound predicted by Toponogov’s question would force the leaf dimension to be zero. Thus positive mixed sectional curvature alone does not imply the Ferus–Adams estimate on closed manifolds. The metrics are not bundle-like. Moreover, the examples can be chosen with mixed sectional curvature arbitrarily close to $1$-pinched, and after normalizing $\max K_{\mathrm{m}\mathrm{i}\mathrm{x}}=1$, the lengths of the leaves remain uniformly bounded. The mechanism is that the positive mixed curvature operator rotates in a parallel normal frame, preventing the scalar Riccati reduction used in the constant-curvature case.