With a candidate like Mr. Rapini running in a state like Connecticut, where the last Republican secretary of state left office in 1995 (and the last one before that left office in 1959), it would be easy enough to mistake his nomination as unimportant.

Richard L. Hasen, a law professor at U.C.L.A. and a leading election law expert, cautioned anyone who might ignore such candidacies. “First of all, just running these races politicizes even further the office of secretary of state,” he told me. Additionally, Mr. Hasen said that having a candidate on a statewide ballot making “constant false claims of massive voter fraud can’t help but create more doubt about election integrity in the minds of a lot of people.”

Although President Biden carried Connecticut by over 20 points in 2020, about 715,000 residents voted for Mr. Trump. If national estimates of support for Mr. Trump’s election lies were to apply in Connecticut, that would mean as many as half a million voters don’t trust elections there already. A statewide candidacy by someone so dedicated to pressing unfounded claims — even if unsuccessful — could, at the least, solidify that election skepticism.

It’s crucial to remember what the office actually means: In Connecticut, as in many states across the country, the secretary of state is the lead elections official. As the commissioner of elections, Connecticut’s secretary of state is responsible for administering its election laws and, under federal law, doing the same for federal elections. Mr. Rapini has made clear that he would use the position to focus on “election security,” as he wrote on the anniversary of the Capitol insurrection. This would be a marked change from the former secretary of state, who was elected three times and focused on “making voter registration and casting a ballot more convenient and obstacle free for every eligible Connecticut citizen.”

Stephanie Thomas, a first-term Connecticut state representative and longtime nonprofit fund-raising and strategy consultant, is the Democratic nominee facing Mr. Rapini in November. “As a nonprofit fund-raiser, the adage used to be if you send someone an email three times, they think they know your organization and they’re more likely to give. So we know that repetition can sometimes prove effective, even if the message is incorrect,” she said in an interview last month. “This type of false narrative just chips away at the fabric of the integrity of our elections, and I think that is just as dangerous as someone in a more reliably red state saying the election was stolen.”

One of the people already in these offices who went through the 2020 election and its aftermath agrees. Jena Griswold, the Colorado secretary of state and the chair of the Democratic Association of Secretaries of State, is running for re-election this year. She told me she worries about a wide range of state and local dangers to voting rights, including threats to election workers, excuses for voting restrictions and local election security breaches, because “local officials embrace conspiracies and become security threats themselves.”