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''Independent of the plausibility or implausibility of the pre-Columbian arrival of Polynesians to the South American Pacific coast, there still would remain the need to explain how these migrants crossed the Andes and ended up in Minas Gerais, Brazil. We feel that such a scenario is too unlikely to be seriously entertained.'

The links in the anthropology blog post which you cite go nowhere useful, but by searching on this quote "In 1808 the Portuguese Crown declared “Just War” (Bellumiustum) against all Indian tribes that did not accept European laws (23). The fierce Botocudo were targeted in such wars and, in consequence, became virtually extinct by the end of the 19th century (24)" I was able to locate and examine this PNAS article.

After dismissing out of hand the notion that pre-Columbian trans-Andean travel could have occurred, the authors eventually arrive at this hilarious speculation about how Polynesian DNA might have ended up in post-Columbian Botocudo Indians (now evidently extinct): The formatting is theirs; bolding is mine.

The last scenario that we wish to assess is the possible arrival of Polynesian haplogroups to Brazil in modern times through the African slave trade from Madagascar, where 20% of the mtDNA lineages belong to the B4a1a1a haplogroup (29). In 1807 Britain outlawed the Atlantic slave trade, making it illegal for British ships to transport slaves. The Royal Navy then began to patrol the waters off West Africa to enforce the so-called “Blockade of Africa.” British cruisers actually succeeded in capturing 169 Brazilian ships in that region in the period 1815–1850 (53). One form of evading the blockade was to switch the trade to the East Coast of Africa, from where ships could take a South Atlantic route to Brazil. Thus, Mozambique, a previously minor source of trade, became in the 19th century a very important port of origin of slaves to Brazil, drawing captives from an ample area that included neighboring Madagascar (53, 54). It has been estimated that between 1817 and 1843 ∼120,000 slaves were brought from that region to Brazil (53), some of them probably going to Minas Gerais, Espirito Santo, and Bahia, to work close to areas occupied by Botocudos. It is known that in these regions, some Botocudo Indians had been pacified during the 19th century and were drafted to work side-by-side with African slaves in plantations (55), an environment potentially conducible to gene flow. Another possibility would be that female slaves from Madagascar living in these regions might have been kidnapped by Botocudo Indians or had run away and find refuge among them, thus creating conditions for introgression of their mtDNA in the Amerindian population. In fact, the kidnapping of a female by Botocudo Indians (Aimores) is a central theme of an 1870 Brazilian opera (“Il Guarany,” composed by Carlos Gomes). Although fanciful, this is perhaps the most likely scenario among those that can be entertained.

In conclusion, we found evidence of Polynesian mtDNA haplogroups in 2 of 14 Botocudo skulls that were studied (30), with independent confirmation of the findings in two separate laboratories. As indicated previously, these findings have bona fide scientific status. We have entertained several possible models to try to explain how these Polynesian sequences were found in individuals from an Amerindian population living in a region in the interior of Brazil. At present, our results do not allow us to accept or definitely reject any of these scenarios. [Including pre-Columbian ones: CE] We hope that further molecular studies will settle the question and will clarify the relevance of our findings for a more complete understanding of pre-Columbian migratory routes of people into the Americas.

Indeed. Further studies seem to be needed. But not by this crew unless they can address pre-Columbian trans-Andean migration scientifically one way or the other, and failing that concede their incomplete knowledge with a touch of humility instead of arrogant bluster.