The only conclusion which is tenable is that there has never been any mixing of humans and Neanderthals, and that any similar genes anybody finds are just artifacts of same or similar low-level components being used by an original designer or designers.
Facts are stubborn things. Sorry.
Once again...
There is terribly little genetic variation in the human race compared to all but one or two other mammals and that is due to some fairly recent genetic bottleneck which reduced the human population to a very small number of individuals, possibly less than fifty or a hundred.
If there ever had been any human/Neanderthal crossbreeding, it would have happened either before or after the bottleneck.
If it happened before the bottleneck, you'd not read these claims of whites and East Asians having Neanderthal genes but Africans not having it; we'd all have them.
If any human group had mixed with Neanderthals AFTER that bottleneck, the variation between that group and all others (humans) would be gigantic, and not minuscule as it actually is.
In other words, the claim of human/Neanderthal crossbreeding cannot pass a test for basic, simple logic. That's just one example of people with PHD degrees who can't deal with logic, there are others.
The basic reality is that whatever may have turned up is just some low-level genetic component which gets switched on and off here and there in the human genome and which Neanderthals may have had as well from original design, it's not anything we ever got FROM Neanderthals.