The dollar has nothing to with what I’m saying.
If Bitcoin forks, producing “Bitcoin A” and “Bitcoin B” chains, the victory of one chain or another will be determined by a small number of people confidently and immediately dumping one to buy more of the other. This will dump the price of one relative to the other, and the “wait and see” folks will get the hint and pile on.
The result is determined by the number of die-hard hodlers of one asset vs the other.