Haven't we known for a while that CHG like ancestry moved into Anatolia and the Levant as time passed, just as "Anatolian farmer" type ancestry moved north and east? That's what the papers have pointed out, as well as some of the amateur bloggers.
Some analyses also have a bit of it in Copper Age people of Europe like Otzi, and in some of the Hungarians, although not all. I think we may see a lot more of it coming at that time, and later in the Bronze Age, once we get some more ancient southern European samples.
There were two big movements out of the Near East. One was from a group centered around Anatolia and the Levant, and one was from the southern Caucasus/Iran. The latter moved into India and southwest into the rest of the Near East, and north as well, ultimately reaching the steppe. The former went into Europe, across North Africa, down into the Arabian peninsula and into Africa and also west and north west into other areas of the Near East. It's a sort of bifurcated "Womb of Nations" to use Dienekes' old term. That ancestry is what binds all those areas together. As pertains to Europe, you have that "Caucasus" type ancestry entering Europe both from the southeast and the east with steppe people.
As for percentages of "Caucasus", or "Iranian farmer", or "southern" or whatever , using admixture programs which use modern people instead of ancient genomes as references is always going to give a distorted picture, because those groups are admixed themselves. Plus, as a counterbalance to admixture, you need to use formal stats. Finally, we need to get access to the papers which have analyzed lots of samples from the Caucasus and surrounding areas for the relevant time periods. Until then, we won't know how much ANE or EHG or whatever was south of the Caucasus at relevant times.
Anyway, that's how I see it.