The juxtaposition of Trump, who has alienated women’s rights activists with his policies, cabinet picks and public posturing, and the premier women’s golf event in the country over which he presides has placed participants this week in an awkward position.

How do they reconcile their ambitions as strong, accomplished women with the expectations thrust upon them by strong, accomplished women whose experience with the president is very different from their own?

And how do the players reconcile the outside outrage over the sexist culture that Trump’s tweets and actions seem to reflect with their personal experience with him as a respectful supporter of women’s golf?

The golfer Lizette Salas, an American of Mexican ancestry, has never met the president, and doesn’t particularly care to. After he was elected, she sent out a tweet expressing her disappointment.

Since Salas was a child, she has dreamed of winning the Open and representing the United States in international competition. With a victory this week, she could set herself up nicely for a third appearance in the Solheim Cup, the top team competition in women’s golf.

Because of her background, Salas has become a target, as her Twitter feed can attest, for groups protesting the president’s ties to golf. She said she sought the advice of her parents on what to do about this Open. They told her she absolutely had to play, and Salas agreed.

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According to L.P.G.A. officials, no players are sitting out the tournament because of where it is being held. The United States Golf Association awarded the Open to the Bedminster club in 2012.

“Women’s golf is in such a critical position right now that in order to make a difference we have to keep playing,” Salas said.

“I get both sides, I honestly do,” she added. “I’ve kind of been put in this position where I’m kind of stuck in the middle. I can take the high road or I could not take the high road and really speak honestly and not make everyone happy. The only win-win for me is I get to play golf.”

The United States Open historically features the most challenging setup in championship golf, but the terrain this week presents an especially tricky test for all the women competing for a piece of the $5 million purse, the largest in L.P.G.A. history. The golfers must also be wary of offending the sport’s elitist white male fan base and the women’s activist groups expected to make their presence felt.

No matter who wins, the victor is likely to be overshadowed by Trump, who is expected to be on the grounds at some point during the weekend rounds.

“I’m sure it’s going to be a circus,” said Juli Inkster, a two-time Open winner and member of the World Golf Hall of Fame, who is part of the Fox Sports broadcasting team for the event.

She added: “But when push comes to shove, it should be about the players and the women’s game. It should be about applauding us instead of bashing us.”

When Kerr won the 2015 Tour Championship for her 18th L.P.G.A. victory, she received two handwritten messages of congratulations. One was from Arnold Palmer, a prolific note writer whose correspondence was often held up as evidence of his grace and generosity. The other was from Trump, who said he was proud of her.

In his life before politics as a real estate mogul, Trump practiced inclusivity, inviting players, including Kerr, to his courses for informal rounds during which he dispensed business advice. For a few years, he hosted the players at his Mar-a-Lago resort in Palm Beach, Fla., during the L.P.G.A. Tour’s season-ending ADT Championship at Trump International Golf Club. In 2006, the tournament became the first women’s event to offer a $1 million first-place prize.

The staunch supporter of women’s golf — and her own career — is the Trump that Kerr knows. She said she did not recognize or condone the behavior of the commander in chief who posts on Twitter disparaging comments about women and who has been accused of harassing women.

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In his tweets, Trump has described a beauty queen as a pig and described Mika Brzezinski, co-host of “Morning Joe” on MSNBC, as “a neurotic and not very bright mess.”

Kerr, 39, knows about cruel comments, having played on her high school’s boys’ golf team in Miami because there was no girls’ squad. In a 1994 article in The South Florida Sun-Sentinel, she dismissed the hostility she encountered by saying: “You know the male ego. Nobody likes to be beaten by a girl.”

Twenty-three years later, Kerr is the highest-earning American female golfer, and yet she is still dealing with subtle sexism that demonstrates some men’s inability to recognize, much less respect, her talent and skill level.

During tournament weeks, she routinely encounters men in her groups at pro-am events who insist on hitting at least one club less than her into the green because they can’t accept that they might hit the same distance, or less, than one of the best female golfers in the world. She registers the surprised looks on her pro-am partners’ faces when they come up short of the green or when she outdrives them. She bites her tongue when they say, “You’re pretty good, you can play,” as if this were a revelation.

“We all say stupid things,” Kerr said.

Politics, sports and gender have become hopelessly tangled this week because of Trump’s connection to golf. While waiting out a rain suspension during the second round at Thornberry Creek last week, Kerr said she wished people could “chill out” and not politicize everything.

She was about to say more, but was distracted by the man who had offered his home near the ninth hole to Kerr, her caddie, her husband, other players and caddies and me as a refuge from an electrical storm that suspended play.

The man and his wife, who were hosting another player for the week, gave Kerr and the others drinks and snacks, but the couple’s kitchen, where Kerr had made herself comfortable, was not a safe harbor. The first warning sign came when I asked Kerr about the women’s groups calling for her and other players to boycott the Open.

The wife was put off by the question and was not mollified by Kerr’s explanation that she had agreed to the interview. The wife left, but her husband stayed. He took out his smartphone and started filming Kerr, who stopped in midsentence when she realized what he was doing. She told him she wasn’t comfortable with the recording and asked him to stop.

He put down his phone but continued to listen to the interview, interrupting again after Kerr’s “chill out” remark to inform her that he didn’t appreciate her attitude, given that he had provided her with food, drinks and shelter. Kerr apologized to the man and said she would leave. On Kerr’s way out, the wife said the couple were very conservative.

Back at the clubhouse, Kerr expressed dismay at what she considered the latest instance of a man’s micro-aggression. “This is why women are so offended by Trump’s comments about women,” she said. “It empowers the bullies and makes women feel small.”

Kerr knows one way to stand tall — to stand above it all. The winner on Sunday can accept the trophy and, through her example, represent strong, empowered women everywhere.

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