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

It is known that the cotangent bundle ${\mathrm{\Omega }}_Y$ 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 ${\mathrm{\Omega }}_{Y\mathrm{\mid }X}$ is stable. We then address some cases where the Picard group increases by restriction.