Jack Dorsey is the CEO of Twitter. He is also the CEO of Square.

That shouldn’t continue any longer, says an investment group, which wants Dorsey to stop running Twitter — or, at the very least, stop running both companies at the same time. And unlike other people who complain about Twitter, these guys might actually be able to do something about it.

Here’s the deal: Elliott Management, an “activist investor” firm, has bought a lot of Twitter shares and wants to replace Twitter CEO Jack Dorsey. It thinks pushing out Dorsey will make Wall Street value Twitter more favorably, which would make the value of Elliott’s Twitter shares rise.

Elliott has acquired about $1 billion worth of Twitter shares, which means it has a sizable stake in Twitter, but nothing close to control of the company. Now, the firm wants to use that stake to replace four Twitter board members with its own nominees, who would then presumably pressure Dorsey to leave — or at least stop running Square, the payments company he founded after he first left Twitter.

Reps for Twitter and Elliott declined to comment. Bloomberg first reported on Elliott’s moves.

There is more to the story, but not a lot more. That’s because Elliott, like any other “activist investor,” thinks it has found a company whose stock should be worth more than it is today. So in many ways it doesn’t matter what Elliott says, or why it’s saying it: As long as Twitter stock goes up, Elliott will be happy.

In this case, Elliott is diverting from the standard activist investor playbook, which is to lay out a critique of many of Twitter’s moves. This approach would also typically involve asking Twitter to slim down by firing people and/or selling off assets as well as calling for new management.

This time, Elliott isn’t bothering with most of that. Its message so far, expressed initially in the Bloomberg piece that announced its plans, is that Twitter will be worth more if it had a full-time CEO, and that the CEO should likely not be Jack Dorsey. Although Dorsey attracts attention for his personality quirks, Elliott’s complaints about his role at Twitter appear to be strictly business-related.

Unlike many tech companies — and some media companies, including the New York Times and Rupert Murdoch’s News Corp and 21st Century Fox — Twitter doesn’t have multiple classes of stock. Companies with those arrangements often give some shareholders — usually founders and their families — extra voting rights and effective control of their companies.

Which means Dorsey is a plausible target for this kind of move. As opposed, to, say Mark Zuckerberg.

Activist investing isn’t a new concept, but Elliott, run by billionaire Paul Singer, has become an increasingly high-profile example of the idea.

A year ago, for instance, Elliott announced that it had bought a lot of eBay stock and then made a series of demands. By September 2019, eBay’s CEO resigned, announcing that he was “not on the same page” as Elliott and the rest of his board. Since then eBay has sold off its StubHub ticketing business and now plans on selling its classifieds business — both moves Elliott had wanted.

Last fall, Elliott also went after AT&T, and the firm got that company to reassess its plans to elevate COO John Stankey to CEO, among other moves. Other Elliott targets include SoftBank, the Japanese conglomerate that owns Sprint and has put billions of dollars into tech companies like Uber and WeWork. Non-US targets have included the governments of Argentina and Peru.

Highly recommended reading: This 2018 New Yorker profile of Singer and Elliott, which should help shoot down facile theories — which you can find on Twitter — about Singer, a prominent Republican donor, going after Twitter for political reasons. Singer and his team are motivated by money above all else. They also don’t want to “buy Twitter.” They want the value of the Twitter shares they bought to increase.

Elliott isn’t the first group to point out that having the same person running two big publicly traded companies at the same time could be a problem. In 2015, Twitter’s board of directors, which was looking for a new CEO, put out a press release that told Dorsey, who was then the interim Twitter CEO, that he couldn’t run Twitter and Square at the same time. Then Twitter’s board changed its mind and gave Dorsey the job, after all.

What’s unclear is why Elliott would go after Dorsey now, since he’s had both jobs for more than four years. Twitter stock has lurched around during that timespan but is a bit higher today than it was a year ago. The company also seems to be making headway on its plan to concentrate on getting Twitter’s user base to spend more time with the service, and to turn that engagement into advertising revenue.

What Twitter doesn’t seem able to do is get a significantly bigger audience to start using Twitter, either again or for the first time. Last year, as its monthly active users numbers stagnated, Twitter stopped reporting those numbers, but they hovered around 300 million at that time. Facebook, by comparison, has 2.5 billion monthly users.

Dorsey does make an inviting target for critics of every variety: Conservatives will complain that his service unfairly silences them. Actual Twitter users frequently complain about the toxic environment they find there, while proponents of democracy worry that the service has been a playground for Russia and other state actors attempting to undermine American elections. It was also somewhat alarming when, last December, Dorsey announced that he would spend up to six months living in Africa this year. Nevertheless, Elliott doesn’t seem focused so much on ousting an eccentric CEO but, more generally, on boosting Twitter’s stock price.

All that said, it’s hard to see what Elliott thinks a Dorsey replacement would do that Dorsey hasn’t. And one contingent that is relatively happy with Dorsey is Twitter’s 4,800 employees, who tend to see him and the company he co-founded in 2006 as one and the same. If Elliott gets its way, it may have a hard time convincing Twitter employees that it made the right move.