“President Xi Jinping clearly reminded us that it is imperative to have a correct understanding of the relationship between ‘one country’ and ‘two systems,’ ” Lam said. “He said one country is like the roots of a tree. For a tree to grow tall and luxuriant, its roots must run deep and strong.”

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This is just what China’s leaders want to hear from the handpicked chief executive in Hong Kong. But the waves of pro-autonomy protests in Hong Kong have severely tested Lam’s political usefulness with Beijing — and underscores a core tension for Hong Kong’s overseers since the territory was returned to Chinese control 22 years ago.

How can you please both Beijing’s hard-line views and Hong Kong’s pro-democracy demands? Or as Kenneth Chan, a former pro-democracy lawmaker and a professor at Hong Kong Baptist University, called it: Hong Kong’s “two masters.”

Lam is now stuck in the middle — a symbol of the competing pressures on Hong Kong at pivotal moment that could shape the political course of the territory for years to come.

Protesters — angered over a proposal to allow extraditions to the mainland — have increasingly called on Lam to step down amid sweeping rallies over the past weeks that have brought millions of people into the streets.

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Chinese authorities, meanwhile, are dismayed by the upheavals and fear that Lam has become so politically damaged that she cannot long longer advance their goals in the semiautonomous territory, according to analysts and others involved in Chinese affairs.

“I would say [Chinese officials] are quite angry,” said Michael Tien, a pro-establishment legislator and business magnate who is also one of the Hong Kong deputies to China’s National People’s Congress.

“They are most furious [at] the [government’s] misjudgment of the acceptance of the bill, and the misjudgment of the reaction of the public,” Tien added.

Steve Tsang, the director of SOAS University of London’s China Institute and an author of several books on Hong Kong’s government, said Lam’s unraveling “is truly extraordinary.”

“She has proven herself to be incompetent. She has proven herself to be a serious embarrassment and she has proven herself to be a liability,” he said. “The Communist Party does not forget and is not very forgiving. She’s finished as chief executive of Hong Kong.”

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There is no sign of Lam bowing out. She tried to ease the protests by pledging to put the extradition bill on hold, and is ready to “accept” the possibility that the bill is likely to die. But she also did not withdraw it.

The huge opposition “obviously touched a raw nerve for people in Beijing,” said Ronny Tong, a legal adviser to Lam who led a delegation to the mainland in May.

Tien, the pro-establishment lawmaker, said that Beijing believed they were “misled from day one that it was just a small opposition to the [extradition bill].”

“The first Sunday protest changed all of that,” he said. “They started thinking, what the hell is going on?”

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Nothing has been easy about running Hong Kong.

Tung Chee-hwa, a shipping tycoon, was the first person to hold the position and resigned in 2005 in part due to mass demonstrations against a sedition legislation that those in Hong Kong feared would end free speech and other rights.

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He was replaced by Donald Tsang, a career civil servant with a fondness for bow ties. The final months of his seven-year tenure were clouded by allegations of kickbacks from business magnates. He was later arrested, found guilty of misconduct and spent 12 months in jail.

Tsang’s successor, C.Y. Leung, rose to international prominence as the scowling, hard-line foil to the student leaders of the 2014 Umbrella Movement, a 79-day sit-in protesting against Beijing’s involvement in the selection of the city’s chief executive. He stunned Hong Kong political watchers by declining to run for the office again in 2016.

Lam finds herself in an even more tenuous position.

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Since her selection as Hong Kong’s leader in 2017, she has overseen the completion or furthering of a number of initiatives that served China’s interests — a high-speed rail connecting Beijing to Hong Kong and a sea-and-tunnel crossing that links Hong Kong to the mainland by road.

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Under her watch, Hong Kong expelled a foreign journalist and charged and jailed leaders of the 2014 protests.

She had been pushing for even more laws that would have pleased Beijing, including one that would criminalize mockery of the Chinese national anthem.

Several Hong Kong government officials say it was Beijing that eventually pushed Lam to back away from the extradition proposals, especially after violence broke out on Hong Kong’s streets.

But the anger has not quelled. Thousands of protesters gathered again outside government offices and police headquarters Friday, calling for Lam to fully withdraw the bill as one of their four demands to the government. Lam has only been willing to put the bill on hold.

Now, even less controversial proposals, such as the national anthem bill, are postponed. Some pro-establishment lawmakers worry their political futures are at risk, and clashed with Lam in a meeting on June 15, before she announced the bill’s suspension.

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“Everybody suffered, everyone is paying a dear price,” Tien said. “I don’t think [Lam] can finish this term unless she completely stays away from controversial issues.”

Shen Dingli, a professor at Fudan University’s Institute of International Studies in Shanghai, said Lam still has some utility to offer Beijing, even as she needs to become better at “predicting the risks of any major undertakings.”

“She is still expected to quiet the turbulence in Hong Kong,” Shen said. “Replacing her has the disadvantage of encouraging the opposition.”

China cannot afford and does not want million-strong political rallies on its doorstep, and has gone to great lengths to ensure that news of the Hong Kong protests does not reach its people.

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Chinese Internet users have posted claims on social media that their government is deleting references to and photos of Hong Kong protests on apps such as WeChat and Weibo. To get around such censorship, users started using more oblique references — referencing a Cantonese pop song, “Queen’s Road East,” that makes reference to the 1997 handover of Hong Kong to China.

But that song was pulled from music streaming services in China. A Washington Post search of the song’s name on the streaming services QQ Music, Xiami, Kugou, and NetEase Cloud Music returned no results Thursday.

In addition, “Boundless Oceans, Vast Skies,” a song by the Hong Kong rock band Beyond that became an unofficial anthem to the Umbrella Movement in 2014, also appeared to have been removed from Chinese music streaming sites.

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Chinese foreign minister Wang Yi on Wednesday, in the first public comments made by a top Chinese official since the protests, said the Chinese central government has “formally expressed our support, understanding and respect” for Lam’s decision to postpone the extradition bill.

But, he claimed, unnamed Western forces are “taking advantage of this issue to stir up trouble.”

“You must withdraw your black hand,” he said. “Hong Kong is China’s domestic affair.”

In Hong Kong, though, distrust of the chief executive position as “political puppets” runs deep, said Chan, the professor at Hong Kong Baptist University.

Even if Lam stepped aside, it would be unlikely that anyone could win back the trust of the Hong Kong public, he added.