The ancestors of pretty much all Europeans could only have come in contact with the Neanderthals in the Middle East so there probably was too much movement and mixing etc. for something like that to survive.

However it’s hypothesized that during the neolithic farmers immigrating from Anatolia/etc. lived alongside the hunter gatherer communities for quite a while. The types of environments preferred by hunter gatherers didn’t have the soil suitable for early agriculture.

But there probably was still quite a but of friction and violence. Also both groups were very different phenotypically, Western Europeans had very dark/black skin and often blue eyes while the middle eastern immigrants had pale skin and dark eyes.