They make it even a bit more clear here in the paper I think:
"The extent to which directional gene-flow occurred from groups related to ancient Anatolia into 317 European HGs predating the spread of farming in Europe has remained controversial. We have provided two 318 further lines of evidence that such gene-flow existed, first using coalescence rates of lineages recently coalesced 319 between Anatolia and HGs. The TCC/TTC mutation rate elevation in all these ancient groups, and its strong 320 correlation to inferred recent shared ancestry with Anatolia, offers complementary support that the shared 321 ancestry detected by Colate indeed reflects recent gene exchange, given the age distribution of samples showing 322 this mutational phenomenon."
So, whether there was gene from European HGs into Anatolia or not, they're arguing there was also gene flow from Anatolia into European HGs BEFORE the arrival of the Anatolian farmers.
They maintain that the signal, the increased TCC to TTC mutation arrived in Europe twice from Anatolia, once in the late Paleolithic, and once with the farmers.
". We do not observe the signal in Neanderthals (Prüfer et al. 2014; 2017) or 267 Denisovans (Meyer et al. 2012), consistent with (Mathieson and Reich 2017). The signal appears already 268 widespread in the Late Upper Paleolithic, as it is carried by Bichon, a 13,700-year-old Western HG, by Sidelkino, a 269 11,000-year-old Eastern HG, by SATP (Satsurblia), a 13,000 year-old Caucasus HG, and Bon002, a 10,000 year-old 270 Anatolian Pre-Pottery individual (Figure 6c, Supplementary Figure 12).
Already 10,000 years ago, the signal appears weaker in Western HGs compared to the Anatolian, who is among the 279 strongest carriers of this signal (similar strength to later Neolithic individuals and present-day Sardinians) (Figure 280 6e), suggesting that the driver of this mutation rate change, which may have been of genetic or environmental 281 nature, was already extinct by the Mesolithic. Eastern HGs have a slightly elevated signal compared to Western HGs. 282 Moreover, the strength of the TCC/TTC signal shows a remarkable correlation with recent coalescence rates to this 283 Anatolian individual (96% using AUC for SGDP non-Africans and 13 high-coverage ancients, 71% using TCC/TTC 284 proportion for ancients)(Figure 6b, d), and does not correlate as well with coalescence rates to any other HG group 285 for whom we have data (88% or 58% with Caucasus HGs (SATP), 83% or 53% with Scandinavian HGs (sf12), 76% 286 or 37% with Eastern HGs (Sidelkino), 73% or 53% with Western HGs (Bichon), where first number uses AUC, 287 second number uses TCC/TTC proportion) (Supplementary Figures 13). We therefore hypothesise that the signal 288 spread through ancestors of this Anatolian individual across Europe before the arrival of farming, and subsequently 289 arrived in Europe for a second time with Neolithic farmers. 290
The genetic relationship among West Eurasian HG groups in the Late Paleolithic is not fully understood and, to the 291 best of our knowledge, current models do not include a clear source group contributing widely across these HG 292 groups, while able to explain the strong correlation to ancestry from Anatolia. One potential source are ancestors 293 of the Dzudzuana, a group inhabiting the Caucasus ~26k years ago (Lazaridis et al. 2018). This group is closely 294 related to ancient Anatolians, and to a lesser extend to Caucasus HGs and may have contributed ancestry to Eastern 295 and Scandinavian HGs before the spread of farming. The Dzudzuana have a pre-LGM common ancestor with 296 Western HGs, including Bichon, however, placing the signal on this common ancestor lineage would not explain 297 their signal strength difference and correlation to shared ancestry with Anatolia. Instead, one possibility is that the 298 signal spread during the Bølling-Allerød interstadial, a brief warming following the last glacial maximum, during perpetuity. "
Huge questions remain. First, what caused this spike in mutation rate change in this triplet. Second, what were the consequences? We know many mutations are deleterious, but for it to have spread so far and wide there must have been mutations which were beneficial, yes?