That's not the question. The question is whether all of the airlines would participate in the fare increase, and if they didn't, would the passengers shun the carriers that raised fares in favor of the carriers that didn't. And I think we all know the answer to that question.
Here is what always happens:
Airline A increases fares by $5. Airlines B, C, and D match the fare increase. But airlines E and F decide that they can snatch some more market share by not matching the fare increase. Expedia shows airlines A, B, C, and D with a fare of $300, and airlines E & F with a fare of $295. The customers all book on airlines E & F, completely filling their flights. Airlines A, B, C, and D are only half full, and their flights operate at a loss. The airlines realize that they can't keep up the fare increase if airlines E and F won't match it, so they are forced to reduce their prices back down to match their competitors in order to not lose market share. Lather, rinse, repeat.
That's the reality of the airline business, folks.