The genetic studies don’t support this.

Ashkenazi are genetically distinct, unlike Sephardic or other Jews. The mitochondrial DNA is traced back to Western Europe, same ancestors of pre-European whites. The haplogroups (paternal DNA) are more diverse. Some lines came from the Caucasus, not the majority.

Further, these gene lines diverge many thousands of years ago, not recent millennium.

https://www.nature.com/articles/ncomms3543