To Fox or not to Fox? That is the question for Democrats running for president in 2020.

In recent weeks, Senator Bernie Sanders and Mayor Pete Buttigieg have made a compelling case for venturing into the lions’ den by holding successful town hall events on Fox News. But senators Elizabeth Warren and Kamala Harris have made a no less compelling argument for shunning Rupert Murdoch’s illiberal network.

Fox News is the top-rated cable news channel in America, with at least one high-profile viewer: Donald Trump himself. Its town halls offer rare exposure to conservative audiences less likely to watch rivals such as CNN and MSNBC. But they could also alienate some Democratic primary voters who regard Fox News as an arch enabler of Trump and all he stands for.

“At a certain point, if you play along with Fox, you don’t look principled or bipartisan, you just look stupid,” the comedian Samantha Bee told viewers of her TBS show. “Nobody is impressed by how Charlie Brown reaches across the aisle to Lucy’s football.”

The debate poses an exquisite dilemma for Democrats with wider implications: whether to try winning back Trump voters or focus on energizing a liberal base; whether to engage with rightwing media or brand it propagandist and destructive; whether the risk of legitimizing a network that carries water for Trump outweighs the risk of deepening divisions in America.

Sanders, an independent senator from Vermont, drew nearly 2.6 million viewers, his biggest TV town hall audience of the primary season, when he appeared on Fox News last month. In one striking moment, the moderator Bret Baier asked if the audience if they would support Sanders’ “Medicare for All” proposal and they erupted in cheers.

Buttigieg, the mayor of South Bend, Indiana, drew an average of 1.1 million viewers and showed the format’s potential to generate viral clips when he condemned two of Fox News’s most popular personalities.

Buttigieg cited “Tucker Carlson saying immigrants make America dirty” and “Laura Ingraham comparing detention centers with children in cages to summer camps”. He went on: “There’s a reason anybody has to swallow hard and think twice before participating in this media ecosystem.

“But I also believe that even though some of those hosts are not always there in good faith, I think a lot of people tune into this network who do it in good faith. And there are a lot of Americans who my party can’t blame if they are ignoring our message, because they will never hear it if we don’t go on and talk about it.”

The Minnesota senator Amy Klobuchar also held a town hall on the channel and is a frequent Fox News guest, contending that it boosts her credentials with voters from both parties. Kirsten Gillibrand, a senator from New York, will have a Fox News town hall on 2 June, which her campaign says reinforces her “taking on the tough fights and winning in red and purple places”.

The New Jersey senator Cory Booker, former housing secretary Julián Castro, former congressman John Delaney, congresswoman Tulsi Gabbard, former Texas congressman Beto O’Rourke and tech entrepreneur Andrew Yang have all said they would be willing to do similar events.

But there is a persuasive case against. Matthew Gertz, a senior fellow at Media Matters for America, a watchdog that frequently criticises Fox News, wrote in a blogpost: “The problem is that the Democratic presidential candidates are allowing themselves to be used by the president’s propaganda network in its ongoing effort to sanitize its brand.”

In March, the Democratic National Committee decided to bar Fox News from hosting one of the party’s 2020 presidential primary debates, with the committee chairman, Tom Perez, contending that the network was not “fair and neutral enough” to do so.

This month, Warren, a senator from Massachusetts, turned down the opportunity of a town hall. “Fox News is a hate-for-profit racket that gives a megaphone to racists and conspiracists – it’s designed to turn us against each other, risking life and death consequences,” she tweeted.

Warren earned widespread praise on social media for an authentic and honest response that appeared to put principle before political calculation. There was also blowback, however. Whoopi Goldberg, co-host of ABC’s The View, respected for its bipartisan audience, responded: “If you can’t face a Fox audience, you can’t face the US. It’s that simple.”

But Harris, a senator from California, has also reportedly rejected an invitation to participate in a Fox News town hall. It remains to seen whether the frontrunner, Joe Biden, will take part.

Analysts suggest there is no one-size-fits-all approach to the conundrum. Michael Cornfield, associate professor of political management at George Washington University in Washington, said of Warren: “Her ‘hard pass’ tweet with its ‘hate-for-profit’ put-down of Fox made more good buzz for her than an appearance would have, whereas Buttigieg needs as much free media exposure as possible, and Sanders relishes confrontation.”

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One factor to wrestle with is the distinction between Fox News journalists, Fox News polemicists and Fox News viewers. Another is the fragmentation of American media, development of echo chambers and breakdown of consensus on basic facts.

Neil Sroka, communications director of the liberal group Democracy for America, said: “There is merit on both sides of the argument. Elizabeth Warren made an incredible case for not doing it. Bernie Sanders, given how capable he was of standing up for his values, did as good a job as you would want.”

What impact the town halls, or lack of them, will ultimately have on the presidential election remains doubtful. The audiences are relatively small: CNN’s town hall with O’Rourke last week averaged just 714,000 viewers, according to Nielsen Media Research.

Sroka, based in Detroit, added: “This is a very inside-baseball discussion. I could see all my Washington friends talking about it, but I’m not sure anyone I talk to here in Michigan was – even those who are active in politics.”