There is now a paper on this topic by Azar, Raina, and Schmalz, the main result is this:
We document a secular increase of deposit account maintenance fees and fee thresholds with a new branch-level dataset, as well as substantial cross-sectional variation in these prices and in deposit rate spreads. We then examine whether variation in bank concentration helps explain the variation in prices. The standard measure of concentration, the HHI, is not correlated with any of the outcome variables. A generalized HHI (GHHI) that captures both common ownership (the degree to which banks are commonly owned by the same investors) and cross-ownership (the extent to which banks own shares in each other) is strongly correlated with higher maintenance fees, fee thresholds, and deposit rate spreads. We use the growth of index funds as a source of exogenous variation to establish a causal link from GHHI to higher prices for banking products.
In other words, if companies are owned by the same pension and mutual funds, why should they compete against each other? Imagine managers given financial incentives for greater stability rather than greater risk-taking, so this does not require a publicly traceable conspiracy.
The first paper on this general question was in fact written by me and Ami Glazer about twenty-five years ago, although we never managed to get it published. Our biggest problem was perhaps the lack of clear evidence at the time. This is the best evidence I have seen so far, although I still file this under “speculative”…
For the pointer I thank Uri Bram.