Extremal-contraction conjecture for slope-unstable Fano tangent bundles

Let $X$ be a Fano manifold, let $n=\dim X$, and define the $(-K_X)$-slope of a subsheaf $\mathcal{E}\subseteq T_X$ by

$$\mu(\mathcal{E})=\frac{c_1(\mathcal{E})\cdot(-K_X)^{n-1}}{\operatorname{rank}(\mathcal{E})}.$$

The tangent bundle has $(-K_X)$-slope $\mu(T_X)=(-K_X)^n/n$. A projective morphism $f:X\to Y$ with connected fibers is a $K_X$-negative extremal contraction when $K_X$ is relatively anti-ample and $f$ is a Mori contraction of an extremal face of the cone of curves. Extremal-contraction conjecture. If $T_X$ is $(-K_X)$-slope unstable, then the relative tangent bundle of a $K_X$-negative extremal contraction destabilizes $T_X$. The conjecture proposes that slope instability of the tangent bundle is detected by an extremal contraction; its resolution status is not specified in the supplied text.