This seems implausible to me, yet there seems to be evidence for it:
Are we kidding ourselves on competition?, by Joshua Gans: ...Consider a situation where there are 10 firms in a market and they compete with one another. Now suppose that all shareholders — say because they are following the dicta of diversification — allocate their wealth in equal proportion across those 10 firms. That means that each owner of the firm — even if there are thousands of these — cares equally about each firm’s profits.
So ask yourself: when those shareholders vote on the composition of boards or the management of the firm, or, importantly how the management of the firm is compensated, are they going to vote for managers who will care only about the profits of the firm they manage or about the profits more broadly? The answer is obvious: they will look to managers who manage in the interest of shareholders and so that means they care about all firm profits and not just the one of their own firm.
In a world where shareholders can get what they want, we won’t have competition in this outcome but, more likely, a collusive outcome. What is more, the firms won’t have to go to all the difficulty of violating antitrust laws to obtain this outcome, they will do it unilaterally. There are no laws against that. ...
Now this isn’t just speculation. Jose Azar, an economist now at Charles River Associates, did his Princeton PhD on this topic. His theory paper is here and it builds on others including Gordon (1990), Hansen and Lott (1995) and O’Brien and Salop (2000). Frank Wolak and I came up with a similar set of issues related to cross-ownership and hedging in electricity markets (for vertical ownership) and verified anti-competitive consequences arising from this. But Azar, along with Martin Schmalz and Isabel Tecu have demonstrated that cross-ownership has anti-competitive impacts on the US airline industry. They find that cross ownership increases US airline prices 3–5%. When they use the event whereby BlackRock acquired Barclays Global Investors (a merger changing the shares of common ownership in airlines), they found such ownership could indicate 10% bumps in pricing with US airline ticket prices rising by 0.6% as a result of that merger alone. ...
The point here is that we cannot really ignore this issue as economists or as policy-makers. We have “known” about it for decades. Now’s the time to take it seriously.
[There's a bit more in the original post.]