It doesn't need to be proven. It's just pure logic applied to economics. You have an injection of capital in the books and this inflates crypto prices.
Also as you mention most of the existing real capital in the books is due to wash trading so a real injection (such as tether) would have an even magnified amplitude of effect.
Did you read the link above? "Tether injections" are characterized by the opposite of a "magnified amplitude of effect."
Yes, it's purely logical that (all else equal) an injection of capital into static supply causes price to increase. But that's no different than buyers depositing to Bitstamp or Coinbase......or Tether. I see no indication that Tether issuance is significant in any way. They are only barely noticeable from a statistical standpoint. I also see no proof they are illegitimate (not backed by real USD) or manipulative.
Your claim looks weak when it's pointed out how small of an effect Tether has. Even if they were trying to prop the market up, they are only a tiny drop in the bucket:
I mean, it's not like we get to see an audit of Coinbase's bank accounts either. They get their share of people shouting "insolvency" too.