Odd-Betti-number conjecture for Fano manifolds with nef tangent bundle

Let $X$ be a Fano manifold with nef tangent bundle $TX$, and let $b_i(X)$ denote its $i$-th Betti number. Odd-Betti-number conjecture.

$$b_{2k+1}(X)=0$$

for every integer $k$ for which the index is defined. The claim is implied by either the cellular-decomposition or diagonal Hodge-number conjecture and hence holds whenever either of those conjectures holds, including the non-negative-bisectional-curvature case. It remains open in general.