I think that the concept of Polynesian types making it across the Pacific to South America is gaining ground. They made it to Easter Island so I would put the probability of this at 100%.
It's also not impossible for random Europeans and Africans to have gotten shipwrecked in the Americas, which would explain small amounts of out of place artifacts and DNA.
My issue is with a trade route existing, let alone imperial administration across the ocean. I don't think that ocean travel would have been reliable enough, think about all the ship wrecks and sailors/colonists dead to disease before the modern era.
The main evidence given is pyramids, even though they are all of different design and that's the most efficient shape to build tall when you have no architectural techniques. Mummification is another one, but that happens by accident all the time.
The most damning fact is that if there was a global community, how come nobody in the Americas were using wheels when the ancient world had a hard on for chariots? Even if the terrain isn't flat enough for chariots, wheelbarrows are a massive labour saving device for a garden.