The normalized tangent bundle classification conjecture for Fano manifolds

Let $X$ be a Fano manifold of Picard number $1$ with dimension at least $3$. The normalized tangent bundle of $X$ is pseudoeffective if and only if $X$ is one of the following varieties: a smooth quadric hypersurface; the Grassmann variety $\operatorname{Gr}(n,2n)$; the Spinor variety $\mathbb{S}_{2n}$; the Lagrangian Grassmann variety $\operatorname{LG}(n,2n)$; or the $27$-dimensional $E_7$-variety $E_7/P_7$. The normalized tangent bundle classification conjecture. This conjecture proposes a classification of Fano manifolds of Picard number $1$ and dimension at least $3$ whose normalized tangent bundles are pseudoeffective. The varieties in the list are known to have pseudoeffective but not big normalized tangent bundles, while the converse classification remains open.