I think it's quite unlikely to be true. This is the best thread about it that I read: https://twitter.com/trvrb/status/1333647437869633537
1) It's probably cross-reactivity from seasonal coronaviruses (as the authors discuss themselves). If I understand correctly, there was only 1 sample in the entire study that tested positive to the most COVID-specific assay (S1-specific Ig). 39 samples tested positive for less specific tests.
2) The raw numbers are not very persuasive (3/519 tested positive in their 'true negative' set of older samples, vs 39/1912 in the set they were evaluating). It's a convenience sample and the authors themselves argue that inferential statistics are inappropriate. But if you ignore the convenience sample and do a Fisher's Exact Test, you get p = 0.02 (not my analysis, see linked Twitter thread). Not very convincing.
3) 39/1912 is about 2%. If you ignore the convenience sample and assume that it is true and representative, where were all the dead people? The hospitals would have been flooded. Also, at least one big testing campaign done in Jan turned up nothing.
Basically, it's an awfully weak result to hang such a major conclusion on. And it's in conflict with a fair amount of other evidence. So I'm quite sceptical and inclined to ignore it, at least until more evidence is provided. It's not impossible but this evidence is too weak to shift my priors meaningfully.