Stability of restrictions of cotangent bundles of irreducible Hermitian symmetric spaces of compact type

It is known that the cotangent bundle $\Omega_Y$ of an irreducible Hermitian symmetric space $Y$ of compact type is stable. Except for a few obvious exceptions, we show that if $X \subset Y$ is a complete intersection such that $Pic(Y) \to Pic(X)$ is surjective, then the restriction $\Omega_{Y|X}$ is stable. We then address some cases where the Picard group increases by restriction.