Diagonal Hodge-number conjecture for Fano manifolds with nef tangent bundle

Let $X$ be a Fano manifold with nef tangent bundle $TX$, and let $h^{p,q}(X)$ denote its Hodge numbers. Diagonal Hodge-number conjecture.

$$h^{p,q}(X)=0\quad\text{if }p\neq q.$$

Equivalently, the Hodge numbers of $X$ are concentrated on the diagonal. The claim follows from the proposed cellular-decomposition conjecture in the source and is therefore known in cases where that conjecture is known, but remains open in general.