Conspiracy theory - Paper Bitcoin

Ok, so I fully admit that this verges on conspiracy, I would like someone with more technical understanding to step in with an explanation but…

If large exchanges, and institutions, and countries, collude, to buy and sell paper Bitcoin, presenting it as real Bitcoin to the public, and opening channels for arbitrage across the market.. couldn’t they set the price? So long as they represent the majority of Bitcoin “holdings”, which as aforementioned is easy because they’re not trading actual Bitcoin for the most part.

Anybody want to shoot this down for me please?

And for clarity, when I say paper Bitcoin, I mean fractional reserves of Bitcoin, I mean that an institution puts on paper that they bought $1.5 billion Bitcoin, but it’s “custodied” in such a way that there’s no actual Bitcoin backing that holding.