They could, but their budgets are their budgets and net tuition is part of their operating budget. So presumably what they would do is increase your first-year tuition so that your estimated four-year tuition would be about the same. Like if it would be 4% higher each year (not compounding) if not fixed, they would make it 6% higher in the first year if fixed, which works out the same over four years.
At a guess, if all the privates could agree to do this at once without violating antitrust laws, they might! However, because they likely can’t do that sort of agreement, then they might see unilaterally doing the one-time 6% increase (or whatever) as too much of a competitive issue.
You might think that doesn’t make much sense, but I think it is plausible as a bunch of people I encounter actually don’t seem to be really thinking about the total four-year cost after possible increases, just the current cost. If enough college consumers are “near-sighted” in this way, then that could in fact be a competitive issue if you do that unilaterally.