Cellular-decomposition conjecture for Fano manifolds with nef tangent bundle

Let $X$ be a Fano manifold, so $K_X^{-1}$ is ample, and suppose that its tangent bundle $TX$ is nef. An algebraic cellular decomposition is a chain of Zariski closed subsets whose successive differences are unions of affine spaces. Cellular-decomposition conjecture. The manifold $X$ has an algebraic cellular decomposition.

This is a weaker alternative to the Campana–Peternell conjecture and would imply the Hodge-theoretic Hopf conjecture. It is known under non-negative bisectional curvature, while its general case remains open.