Campana–Peternell width-decreasing conjecture for CP-manifolds

A CP-manifold is a Fano manifold with nef tangent bundle. For a CP-manifold $X$ of Picard number $n$, choose minimal rational curves $\Gamma_i$ associated with its elementary extremal rays and define its width by

$$\tau(X):=\sum_{i=1}^n(-K_X\cdot\Gamma_i-2).$$

A CP-manifold is of flag type when all its elementary contractions are smooth $\mathbb{P}^1$-fibrations; equivalently, its width is zero.

Width-decreasing conjecture. Let $X$ be a CP-manifold which is not a product of positive-dimensional varieties. If $\tau(X)>0$, then there exists a surjective morphism $f:X'\to X$ from a CP-manifold $X'$, which is not a product of positive-dimensional varieties, such that $\tau(X')<\tau(X)$.

This property would complete the proposed strategy for the Campana–Peternell Conjecture: the width would decrease until one reaches a flag-type manifold, which is a complete flag manifold by the theorem cited in the source. The source does not report a resolution of this conjecture.