I don't subscribe to their claim. It may be running through generations but clearly it couldn't be traced back to ancient Israel.
The village of Piq'in in Israel has an unbroken chain of Jewish families that lived their for over 2400 hundred years.There is still 1 family residing their i think their name is zannati
I know about Pq'in. I visited there several times. The last Zinnati living there, is a nice old woman. She keeps the keys for the synagogue and open it for visitors. The story of the Sofer family, that they lived in Israel and in Yemen simultaneously sounds unreasonable to me. In fact, for hundreds of years there was very limited contact between Yemen and Eretz Israel or the world Jewry anywhere. Which is why the Ymenite Jewry is distinctly different from both Sfarad and Ashkenaz.