Iskovskikh's stability conjecture for tangent bundles of Fano manifolds

Let $X$ be a Fano manifold, meaning a smooth projective variety whose anti-canonical divisor $-K_X$ is ample. Let $\rho_X$ be the Picard number of $X$, and let $\Theta_X$ denote its tangent bundle. Assume that

$$\rho_X=1.$$

Iskovskikh's stability conjecture. The tangent bundle $\Theta_X$ is (semi)stable.

The conjecture connects stability of tangent bundles with the geometry of Fano manifolds and was attributed to Iskovskikh. It has been confirmed for Fano manifolds of index one, smooth complete intersections in projective space, and further classes; the paper's abstract states that the conjecture is disproved by examples with unstable tangent bundles.