Their main point clearly is that "By analyzing this data, we find that the Anatolian hunter-gatherers are genetically distinct20 from other reported late Pleistocene populations and thus represent a previously undescribedpopulation. We reveal that Neolithic Anatolian populations derive a large fraction of theirancestry from the Epipaleolithic Anatolian population, suggesting farming was adopted locallyby the hunter-gatherers of central Anatolia."
More specifically:
"Accordingly, wefind an adequate two-way admixture model using qpAdm12 (χ2p = 0.158), in which AHG derivesaround half of his ancestry from a Neolithic Levantine-related gene pool (48.0 ± 4.5 %; estimate± 1 SE) and the rest from the WHG-related one (tables S4 and S5). These results support a late15 Pleistocene presence of both ancestries in a mixed form in central Anatolia. Notably, the geneticconnection with the Levant predates the advent of farming in this region by at least fivemillennia and potentially correlates with evidence of human interactions between centralAnatolia and the Levant during the Epipalaeolithic13.
In turn, AAF is slightly shifted upwards compared to AHG in the PCA, to the direction20 where ancient and modern Caucasus and Iranian groups are located. Likewise, when comparedto AHG by D(AAF, AHG; test, Mbuti), the AAF early farmers show extra affinity with earlyHolocene populations from Iran or Caucasus and with present-day South Asians, who have alsobeen genetically linked with Iranian/Caucasus ancestry14, 15".A mixture of AHG and Neolithic Iranians provides a good fit to AAF in our qpAdm modeling(χ2p = 0.296), in which they derive most of their ancestry (89.7 ± 3.9 %) from a populationrelated to AHG (tables S4 and S6). This suggests a long-term genetic stability in central Anatoliaover five millennia despite changes in climate and subsistence strategy.So, some small influence From Iran or Caucasus by the Holocene, but then the direction of gene flow changes.
"In contrast, we find that the later ACF individuals share more alleles with the earlyHolocene Levantines than AAF do, as shown by positive D(ACF, AAF; Natufian/Levant_N,Mbuti) ≥ 3.84 SE (Fig. 2B, fig. S3 and data table S3). Ancient Iran/Caucasus populations andcontemporary South Asians do not share more alleles with ACF (|D| < 3.3 SE). Likewise,qpAdm modeling suggests that the AAF gene pool still constitutes more than 3/4 of the ancestry15 of ACF 2,000 years later (78.7 ± 3.5 %; tables S4 and S7) with additional ancestry well modeledby the Neolithic Levantines (χ2p = 0.115) but not by the Neolithic Iranians."
"In turn, Levantineearly farmers (Levant_Neol) that are temporally intermediate between AAF and ACF could be20 modeled as a two-way mixture of Natufians and AHG or AAF (18.2 ± 6.4 % AHG or 21.3 ± 6.3% AAF ancestry; tables S4 and S8 and data table S4), confirming previous reports of anAnatolian-like ancestry contributing to the Levantine Neolithic gene pool."
So, as they say: "detected gene flows support a reciprocal genetic exchange between the Levant and Anatoliaduring the early stages of the transition to farming".