Chaff faced ware (4300-3500bc) was stretched from Cilicia to Caspian Sea. This cultural horizon is the common denominator of Minoan Y DNA and Maykop Y DNA. Their common ancestor.
One group moved to north from South Caucasus where a variant of this culture is known as Leila Tepe. In the north they participated to the formation of Maykop culture. While another group moved to Crete and Greek islands..From Aintab-Urfa region. It's also possible that CFW related groups moved to Central Asia. This is the reason why we have similar haplotypes in both places. T1a3 and L2 both in Maykop and Greece. With more sampling from Maykop we will find more common haplotypes.
We have some aDNA from CFW period. It's Areni from Armenia and Azerbaijan LC. Both can be used for the modeling of Maykop in qpadm. See qpadm models in the comments.
As for Sioni-Adablur culture (5300-4300/4000bc) they were a different group who were adding grit to the pottery. We don't have ancient DNA from that period and sites.
The source
1. Sagona A. In: The Archaeology of the Caucasus: From Earliest Settlements to the Iron Age. Cambridge World Archaeology. Cambridge University Press; 2017:v-v.