No, the banks get underwriting fees as a % of the offer, so they are incented for a higher price as well.
The 'winners' are the folks who get to buy from the underwriting bank and that's opaque: it could be 'the same banks clients', and so they get some major favour points for those allocations, and their own private wealth managers would ostensibly get a cut, but that's quasi insider stuff.
The banks 'working with the IPO company' lose money with a price that's too low.
Even if you were to say 'oh, the other parts of the banks are working for some action' the answer to that is, well, bankers are 'very me first' and I don't care who it is, the underwriting team wants to make as much money as they can, it's their personal bonus, and they're going to maximize the price. They don't care about the 'private wealth management teams' bonuses, they care about their own.
AirBnB I guarantee literally had some of the world's best bankers on the case.
There's just a lot of variation in very frothy markets and it's hard to estimate demand from far flung corners.