If this is true, it doesn’t matter.
Being Jewish is being a member of G-d’s Covenant at Mount Sinai. Over the centuries, some women and men said, in effect, “Count me in,” and so they became members of the Covenant by observing its specifics and generalities.
If a genetic study reaches the conclusion that many European women converted, it is consistent with what we know of Roman history.
A history of Jews in the Roman Empire
Until the end of the first century CE, Judaism was the only monotheistic religion. Non-Jews worshiped a pantheon of gods. Some worshiped the god of their own city-state but not the god of any other city-state.
People of the Roman Empire were supposed to worship a statue of the emperor as well as any other god that they chose. The Emperor Hadrian was especially diligent in dedicating statues of himself all over the eastern part of the empire. His was particularly fond of the Greek culture and “Greek love” — pederasty.
We find evidence that non-Jews spent time around and in synagogues. These people have been called G-dfearers and Phylo-Jews — Jew lovers. It’s not so much that they liked Jews as such. They liked the religion of Jews. These G-dfearers were a pool of prospective converts. The “fly in the ointment,” the drawback, for men to convert was that they would be undergoing circumcision. I’m under the impression that adult circumcision is painful and takes a while to heal. It is speculated that not many adult men converted, but women did.
Some of the female converts probably married Jewish men. Perhaps some Phylo-Jews circumcised their sons in hope that these boys would convert. Who knows? And no one knows the numbers.
Perhaps as population genetics improve, studies of Jewish genetics may provide fine grain statistics for reconstructing theories.
Regardless of some of the previous details and the associated speculation, historians have long suggested that about ten percent of the population of the Roman Empire was Jewish. Recent reevaluations are suggesting a smaller Jewish population. But, imagine that the U.S. had a Jewish population of 38 million, or that 19 million Jews lived in the U.S. These figures are about 10 percent 5 percent respectively.
So now back to Rome. However many Jews lived within the Roman Empire, they were not evenly spread out. The largest concentration of Jews seems to have been on the Italian peninsula and in Rome itself. Many Jews did not relocate voluntarily. When the Roman Legions destroyed the Holy Temple in Jerusalem (69 CE), and when they destroyed the city of Jerusalem (about 135 CE), Jewish captives were taken to slave markets. As defeated captives, they were paraded through the streets of Rome.
After Christianity became the official religion of of the Roman Empire (4th century CE), conversion to Judaism was punishable by death. However many European women (and men) had affiliated with the Jewish people before this, no more converted.
In time, Roman Jews (from Rome and Italy) found opportunities northward, over the Alps into the Rhine River valley and also in Roman Gaul. These Jews were the forebears of Ashkenazi Jews. A particular regional identity did not start to develop, though, until Charlemagne’s power reached from the Elbe River in today’s Germany, south and west to the Pyrenees Mountains, and southward to encompass all of Italy. This is the core of the Western Roman Empire, and this is the spiritual birthplace of Ashkenazi customs and their heritage of studying the Bible and the Talmud..
Jews in this region primarily had spoken Vulgar Latin — the language of their neighbors. Charlemagne’s Empire, the Carolingian Empire, brought Germanic speakers into the region from the east and northeast. The legal and language frontier along the Rhine River supplied the core population of Ashkenazi Jews.
Ashkenaz — Germanic Europe
I’m not aware of the reference ‘Ashkenazi’ appearing before the year 1000 or so. The earliest usage referred to the Jews of the Rhine River region and the locality of Charlemagne’s preferred seat of power in Aix-la-Chapelle/Aachen, only a short distance west of the Rhine. It is no surprise that Ashkenazi Jews would come to speak Yiddish — Jewish German.
This first usage came about when two distinguished Jewish families from Italy crossed the Alps and settled near the Carolingian seat of power around the year 900 (or so).
We find a dichotomy here, though. The spiritual roots of Ashkenaz were in two Jewish cultural and scholarship areas. The guiding spirit of Ashkenazi Jews came from Italy. For the most part, Italian Jews remained in Italy, though.
The much of intellectual ferment of Ashkenaz came from the Torah academies in southern France. Again, for the most part, these Jews remained near the Mediterranean.
Work in progress -- nearing conclusion.