Weak Campana–Peternell conjecture on bigness of nef tangent bundles

Let $F$ be a Fano manifold with nef tangent bundle $\mathcal{T}_F$.

Weak Campana–Peternell conjecture. If the tangent bundle of a Fano manifold is nef, then it is also big.

This is recorded as a weak version of the Campana–Peternell conjecture. The source notes that global generation implies bigness and that bigness implies semiampleness, but the asserted implication is not known in full generality.