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