The move from Musk came after he posted an unscientific poll on his personal Twitter account that concluded Friday night with 59% of participants voting in favor of immediately restoring the accounts.
Twitter bans accounts of several journalists from top news organizations without explanation
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Musk had on Thursday banned CNN's Donie O'Sullivan, The New York Times' Ryan Mac, and The Washington Post's Drew Harwell. Independent progressive journalist Aaron Rupar, former MSNBC host Keith Olbermann, and Insider columnist Linette Lopez were also banned.
"The people have spoken," Musk wrote Friday night after his poll, pledging to restore the accounts he had falsely accused of sharing his "exact real-time" location.
But while the accounts were made publicly viewable on Saturday, the journalists were restricted from posting until they removed the tweets Musk had claimed violated Twitter's rules.
In the past, Twitter had required the removal of violative tweets for users to regain access to their accounts, but the journalists in this case strongly dispute that their posts violated Twitter rules.
O'Sullivan and Harwell both told CNN on Saturday morning that they had not agreed to delete the tweets and instead selected an option to appeal the decision.
"It's journalism," Harwell wrote in his appeal, a copy of which was provided to CNN. Harwell added that his tweet did not include a "link to anyone's private information."
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Rupar told CNN that he had ultimately decided to simply remove the tweet and move on from the episode, though he described the whole affair as "kinda (sic) absurd obviously."
It was not clear what Mac had chosen to do.
The accounts of Olbermann and Lopez notably remained banned and had still not even been made publicly available by late Saturday morning.
Musk had falsely claimed on Thursday that the journalists had violated Twitter's new "doxxing" policy by sharing his "exact real-time" location, amounting to what he described as "assassination coordinates."
The suspension of the journalists had been met with swift condemnation by news organizations, the American Civil Liberties Union, United Nations, Democratic members of Congress and others.
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The move marked a significant attempt by Musk, a self-described free speech absolutist, to wield his unilateral authority over the platform to censor the press.
A CNN spokesperson previously said on Thursday that the network had asked Twitter for an explanation over O'Sullivan's suspension and it would "reevaluate our relationship based on that response."
Shortly before his suspension, O'Sullivan tweeted that Twitter had suspended the account of an emerging competitive social media service, Mastodon, which has allowed the continued posting of @ElonJet, an account that posts the updated location of Musk's private jet.
Other reporters suspended Thursday had also recently written about the plane-tracking account, which Twitter permanently suspended the day before as it rolled out a new policy prohibiting the sharing of live location data.
The move to ban the jet-tracking account marked a sharp reversal of Musk's vow to leave the account online as part of his "commitment to free speech."