A new paper, Nina Marchi, et al. "The genomic origins of the world's first farmers" Cell (2022) argues based upon analysis of a medium sized sample of ancient hunter-gatherer DNA that early European farmers and early Iranian/Caucasian farmers did share a common genetic hunter-gatherer origins and that two rounds of Western hunter-gatherer introgression of ca. 14-15% each into the Mesolithic hunter-gatherer gene pool ancestral to early European farmers, accounts for the differences between them and early Iranian farmers.
Bernard's blog explains that:
The results show that contrary to previous studies, hunter-gatherers in the Caucasus and farmers in Europe and Anatolia are all descended from the central meta population. This metapopulation received about 14,200 years ago a contribution (14%) from the western metapopulation. The ancestors of the farmers of Iran did not receive this western contribution. The central and eastern metapopulations diverged about 15,800 years ago.The ancestors of the European and Anatolian farmers received a second gene flow (15%) from the western meta population about 12,900 years ago. The ancestors of the Caucasian hunter-gatherers did not receive this second contribution. The ancestors of European and Anatolian farmers then underwent a major genetic drift between 12,900 and 9,100 years ago due to a strong reduction in their population.