The semistability conjecture for tangent bundles of Fano manifolds

Let $X$ be a Fano manifold with Picard number one. The semistability conjecture. The tangent bundle $TX$ is semistable. This conjecture was disproved by Kanemitsu, who found smooth horospherical counterexamples; the source specifically describes the non-semistable types $(B_n,\varpi_{n-1},\varpi_n)$ for $n\geq 4$ and $(F_4,\varpi_3,\varpi_2)$.