This is a follow-up to this question, which asks for an example of a non-regular language $L$ which satisfies the following condition, intersection resilience:
If $L'$ is context-free then so is $L \cap L'$.
We know that all regular languages are intersection-resilient, and clearly every intersection-resilient language is context-free. Ginsburg and Spanier showed in 1964, in their paper Bounded ALGOL-like languages, that every context-free subset of $x^*y^*$ is intersection-resilient (Theorem 6.1, Corollary 2).
Which languages are intersection-resilient?