I'm not qualified to answer your question ~specifically~ (although, on the same subject matter); but I would like to point out that IF it's true that only a tiny number ("1-4 percent of modern European") people carry Neanderthal genes, that can only mean Homo Sapiens sapiens didn't evolve from them; because if it did, then practically everyone would carry some of those same genes.
Instead, modern Humans carry an astonishingly vast & varied mosaic of 'junk DNA'. I saw another article link recently (on this forum, I believe) which reports the recent discovery that 'junk DNA' may be used to distinguish Human vs Apes' genetic inheritance.
For example, some modern Humans carry a few of the same DNA mutations as Chimps; some carry DNA that Gorillas also share; some carry DNA found in Rhesus monkeys; & probably most of us carry combinations of the three (or others). Humans & other primates also have overlaps in blood groups & types.
Naturally we're all primates (although as a species Humans are obviously at the top of the IQ scale); so we should expect certain kinds of DNA is shared among all primates. Similarly, I'm pretty sure other kinds of DNA are shared among all Mammals.
Specialists have developed criteria to distinguish hybrid genes, however; & I've seen it reported scientifically, that Neanderthals were a hybrid species. I believe the subject of Human hybridization deserves more attention (though it seems to be a touchy one for a lot of people, unfortunately).