Language whose intersection with a CFL is always a CFL (2)

Question

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?

Answer 1

Just a few notes (I'll try to add the proofs later).

Intersection-resilience seems closed under homomorphism.

A necessary condition for intersection-resilience is that for all homomorphisms $\varphi : \Sigma \to \Sigma \cup\ \{\epsilon \}$; $\varphi(L) \cap x^*y^*x^*$ must be a regular language (where $x$ and $y$ are any two distinct symbols of $\Sigma$).

So the result of Ginsburg and Spanier seems "tight" (if we use the same homomorphism trick).