In the aftermath of Tuesday’s caucuses, the haunting cry went up among Minnesota politicos: Beware the curse of the straw poll.

How important is the straw poll?

“Well, just ask Gov. Sullivan and Gov. Seifert,” state Sen. Dave Thompson, R-Lakeville, said, followed by a long stream of laughter. Thompson, a gubernatorial candidate, came in a close second in Tuesday’s nonbinding Republican poll.

In 2002, Republican gubernatorial candidate Brian Sullivan bested then-House Majority Leader Tim Pawlenty in the caucus straw poll. Pawlenty won the GOP endorsement and then the governor’s race.

In 2010, candidate Marty Seifert handily won the GOP’s caucus straw poll, but Tom Emmer won the party’s endorsement. That year, Democrats also had a gubernatorial straw poll and Minneapolis Mayor R.T. Rybak narrowly won it. He later lost the DFL endorsement to then-House Speaker Margaret Anderson Kelliher, who then lost a primary to now-Gov. Mark Dayton.

Former House Minority Leader Seifert made a case that he can shake off the haunting this time. “Anything can happen,” he said last week. Running for governor again, Seifert, of Marshall, won the Republican straw poll with 29 percent of the vote last week, doing particularly well outstate.

“Mark Dayton should not have won the primary, should not have won the general election, was locked out physically, literally, from his own convention four years ago,” Seifert said. “Various things happen in Minnesota that break the mold, and we’re here to break the mold this year on the track record of that.”

Tuesday’s caucus attendees not only voted in the straw poll but also elected delegates to the next level of political gatherings. Eventually, their leaders will select the Republican candidate who will carry the party endorsement into the August primary. Candidates can run in a primary without endorsement, but party resources and many activists will back only the endorsee through a primary.

Despite that potential power, only a tiny portion of Minnesotans turn out for caucuses. On the Republican side, which features a U.S. Senate contest for GOP primacy as well as the governor’s race, only 13,000 people voted in the straw poll.

And they won’t necessarily be the ones who move on to the next level.

“Of those who voted in the caucus straw poll, only 15 [percent] will go on to become delegates at the state convention. We simply need to out-organize and outwork the other candidates as we go through the process of choosing that 15 [percent],” gubernatorial candidate Jeff Johnson said in a Facebook message. He captured 17 percent in the straw poll, a distant third to Seifert and Thompson.

The history, and the low turnout from Tuesday’s caucuses, gives fuel to critics of the caucus system. Their argument largely goes like this: So few people turn out for caucuses that they are unrepresentative of most voters’ thoughts and concerns. Why waste resources trying to capture the interest of those activists, who tend to be the more extreme on both ends of the spectrum?

But, for now, the caucuses continue to be part of the game.

“It’s a barometer, but it is by no means definitive,” said state Rep. Kurt Zellers, a Republican from Maple Grove who is running for governor. He came in behind “undecided” in Tuesday’s straw poll, pulling just 8 percent. Candidate Scott ­Honour, whose campaign said “caucus straw poll has never been part of our strategy or path to victory,” got 9 percent.

Although Zellers did not do well Tuesday, he said he still respects the caucus process.

“I’ve been endorsed six times. I’ve been going to caucuses for 24 years,” he said. Like Honour, he has made clear he would run in a primary no matter whom delegates endorse at their convention in May. “We have an ‘all of the above’ strategy.”

Thompson, who did particularly well in suburban areas on Tuesday, has said he’ll stay in the race only if he captures convention backing. That made the caucus results especially important for him. “You can’t win the election on caucus night,” he said. “But you can lose it.”