I see... I suspected that AJs, both German/French AJs on one hand, and Hungarian/Polish/Russian AJs are pretty much the same people. But I don't think SJs went through a bottleneck, did they?
I guess it makes sense that the gene flow went both ways, probably due to trade and such during the bronze age, and during the Hellenistic period also due to trade, but also because many Jews migrated to the Greek isles, specifically Hellenistic Jews (the first Jews recorded in Europe during the early part of the 3rd century B.C.E all had Greek names and were recorded in Greek territories), but also because, as you've said before, some Greek women may have married Jewish men, it would have been easier back then because Judaism passed from the dad back then, and women didn't and don't have to have a Bris to convert. I suppose that the fact that AJs plot between Cypriots and Greeks isn't due to gene flow from Italians, but gene flow from and to Greeks, as your map clearly shows.
But I still am not entirely sure how there's a gene flow from Sardinians and Basques, could it be from a common pre Islamic east Mediterranean ancestry?
In conclusion, due to the lack of WHG among AJs, yet with AJs still plotting in the gap between Europe and the near east, we can expect to have a gene flow from a European population with very little to no WHG ancestry, modern mainland Greeks only have around 6% WHG ancestry, imagine how much they had before the Slavic/Germanic migrations, very little, and the fact that there seems to be gene flow from them, suggests that the European admixture that pulled AJs from Cypriots to Sicilians and Maltese, may indeed be pre Slavic Greeks, and perhaps as you've said, even before the Hellenistic period, during the Bronze age, considering the fact that the Philistines probably came from the Aegean. This really answers a lot of questions, considering the fact that Sicilians came under Hellenistic influence, and modern Maltese descend from Sicilian settlers. Also let's not forget that it's not like there's no mention of Hellenistic influence on Jews during the Hellenistic period, quite the contrary! :)
P.S I'm pretty sure you've answered that, and I know it'll be hair splitting, but AJs, while technically having 0% like Sicilians and Maltese, might have some via the EEF ancestry, also let's not forget that while AJs have 0, they don't have negative, as Cypriots do.
P.P.S Yeah, I suppose that the Cossack rape didn't have an overwhelming influence on AJs, which would basically make them a half Slavic half east Mediterranean people, if that would happen, AJs would plot between Cypriots and Ukrainians, and would have very visible WHG ancestry, indeed, if AJs were a half German half east Mediterranean people they would have visible WHG ancestry. I reckon that there was indeed more gene flow from AJs to the larger Christian Slavic population than the other way around, many Jews converted to eastern Orthodoxy in order to leave the pale of settlement and receive better rights, some did it to advance in the military, or some were kidnapped and thus assimilated into the larger society, but there were some cases, I know of a couple of Russian Jews that look like Cossacks, and who are aware of some Cossack ancestry.
P.P.P.S Don't forget that there were Greek colonies in Anatolia, as well as many Hellenistic cities, this could have genetically influenced the Anatolians as well.