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It has not been a good year for the Mark Zuckerberg name, as Facebook has fallen deep into a series of privacy scandals over allowing millions of users’ personal data to be improperly exploited.

And in San Francisco it’s become a particular flash point. Here, the Zuckerberg name is visible on one of the city’s most iconic institutions and one where privacy is part of the compact: the 146-year-old general hospital. After a record-setting $75 million gift from Mr. Zuckerberg and his wife, the pediatrician Priscilla Chan, in 2015, the general hospital was renamed Priscilla Chan and Mark Zuckerberg San Francisco General Hospital and Trauma Center.

Now, in light of Facebook’s recent controversies, a group of nurses are saying the name makes patients wary. And they have renewed their efforts to have it removed.

“We are in charge of keeping our most vulnerable people private and protected,” said Heather Ali, who works at the hospital in nursing administration. “Now people wonder, ‘How much is my privacy protected at a hospital with that name on it?’”