Jack Dorsey may want to rethink his dual role as CEO of social network site Twitter  (TWTR) and mobile payments startup Square  (SQ) .

Both companies are trading at or below their initial public offering price. If Twitter's share price keeps dropping it could soon become the target of a tech-focused activist hedge fund urging a sale and Dorsey's ouster.

Dorsey, who founded both companies, has recently announced some changes, with four top VPs on its executive team exiting the company and one new hire, a new chief marketing officer, brought on board Tuesday.

Insurgent investors may want to give Dorsey's new team -- including a soon to be announced "high-profile media personality" on its board -- some time before launching an activist campaign. However, without a major turnaround soon, Dorsey's position as CEO of two major publicly traded companies becomes a significant vulnerability that could attract an activist.

"The biggest single problem that shareholders have with Dorsey is that he is not able to devote his full time to Twitter and the share price slide suggests he should spend his full time with Twitter," said Wedbush Securities Inc. analyst Michael Pachter.

To be CEO of two publicly traded companies is too much, said Eric Jackson, managing director of activist fund SpringOwl Asset Management and a columnist for Real Money and TheStreet. This is especially true when one of the firms, Twitter, is in the midst of a turnaround.

In tech, dual CEO roles are rare, with Elon Musk, currently CEO of Tesla Motors (TSLA) and privately held SpaceX, formally known as Space Exploration Technology Corp., reportedly recommending against taking on two chief executive positions. 

In addition, a large minority of shareholders have already expressed their disappointment with Twitter at the ballot box. Roughly 31% of the voting shares opposed the re-election of Twitter director David Rosenblatt, an ex-Google and DoubleClick ad executive, at the company's annual meeting in June. The vote was substantial and likely directed at broader concerns about whether Twitter really needs two ex-Google directors -- Omid Kordestani, Twitter's chairman, previously served as chief business officer at the online search giant.

An activist could also seek to curry favor with Twitter's institutional investor base by pointing to its poor governance ranking. The social network company received the lowest rating possible, a 10 out of 10, for governance by Institutional Shareholder Services. According to a report obtained by The Deal, Twitter received a red flag for a "pay for performance" misalignment based on the proxy advisory firm's qualitative review. It also issued a red flag around Rosenblatt's poor election result.

Wedbush's Pachter said he is not surprised about the ISS concerns around compensation. He calculates Twitter's roughly 4000 employees receive on-average roughly $175,000 each in annual stock-based compensation. "We all tolerate it when the stock price goes up," Pachter said. "But not so much when the stock price is down."

However, any activist that would launch a campaign at Twitter would face significant obstacles. For one thing, Twitter's board is classified, which means the activist could, at best, succeed at installing a minority slate of dissident directors who would have some influence but couldn't remove Dorsey on their own. In addition, the annual meeting is the only place one can elect directors at Twitter so they would need to nominate director candidates by a March 6 deadline.

In addition, roughly 15% of Twitter's outstanding shares are owned by insiders, including 3.7% stake owned by Dorsey, according to the company's 2015 proxy statement.

Also, SpringOwl's Jackson argues most activist funds don't have enough expertise in the area of social networking companies or how to fix a user growth problem at Twitter. "They would probably worry about looking foolish by taking a position in Twitter and coming out with a bunch of prescriptions about how to improve a consumer-facing app," Jackson said.

While many activists seek to break up companies, Twitter can't really be broken up. Pachter asserts that its Tweetdeck and Vine subsidiaries could not survive as standalone businesses.

Nevertheless, if the stock price drops low enough, Jackson said an activist could seek to gin up interest in having Twitter sold even as he concedes that Dorsey is unlikely to want to sell the business for anything below its $26 a share IPO price. Twitter's stock price has dropped from roughly $31 a share in October to the current $16.77.

According to Jack Mohr, director of research for Jim Cramer's charitable trust, Action Alerts PLUS, this could be exactly what happens. Twitter is an AAP holding.

"Much has been made about the immense, unlocked potential that the microblogging company harbors, and many believe that it could draw upon the scope, scale and expertise of a successful social media or news conglomerate to unlock its inherent value and turn the struggling company around," Mohr said Thursday. "Takeover rumors around [Twitter] have been abundant since last April, but the subsequent 70% valuation compression since the earliest report makes the case for being acquired more reasonable."

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Potential buyers include Alphabet  (GOOGL) , Facebook (FB) , Alibaba (BABA) , Microsoft (MSFT) and Yahoo! (YHOO) . Rupert Murdoch's News Corp  (NWSA) last week denied rumors it is interested in buying Twitter. 

For some, the only real strategic fit for Twitter would be Facebook. Pachter suggests that embattled Yahoo! CEO Marissa Mayer could seek to buy Twitter as part of an effort to reinvigorate its brand and help it fend off its own activists. Starboard Value's Jeff Smith has threatened a change-of-control proxy fight at Yahoo if it didn't replace management and quickly explore a sale.

"Facebook is missing real time news and Twitter allows people to follow things they are interested in rather than people they know," he said. "If Marissa Mayer wants to keep her job she has to do something and buying Twitter could be it."

And how would the activist launch their campaign? An insurgency could effectively take off with a tweet - following in the footsteps of Carl Icahn, who began his Apple (APPL) insurgency with one. It seems like that would be the most Twitter-friendly approach, with a quick securities filing and hashtag alert, #sellTwitter, to follow.

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