Let V=W[g], where g is P-generic over W for some poset P in W. Let F be a V-extender with critical point κ such that P ∈ V_{κ}^{W}. If the support of F is sufficiently closed, say strength(F)=length(F)=λ for some inaccessible cardinal λ>κ, then F ∩W ∈ W. To the best of my knowledge, this is due do Hamkins-Woodin; the argument for it which I have in mind is the one written up in https://ivv5hpp.uni-muenster.de/u/rds/VM2_4.pdf .

Can this be proven in more generality? I.e., what if we don't assume ult(V;F) to be Card(P)-closed, is it still true that F ∩W ∈ W? Or is there a counterexample?

And can one even drop the hypothesis that F be a **total** V-extender, i.e., that it acts on some transitive model contained in V rather than on all of V?