Specifically, they asked the party to make caucuses more accessible and the more contentious change require candidates to square off in a primary if they didn't get at least 70 percent of the delegate vote at convention, an increase from the current 60 percent threshold.
The group contends that would lead to more primaries, more voter involvement and higher turnout.
But the Republican State Central Committee soundly defeated a proposal to raise the threshold to 70 percent and also rejected raising the threshold to two-thirds of the delegate pool.
"Don't be blackmailed," urged Don Guymon, of Davis County. "Let's do what's best for the Republican Party."
Peter Cannon, also from Davis County, argued America is a republic that relies on informed members representing them, not a pure democracy.
"Our Founding Fathers did not like a democracy and we don't like it, either," Cannon said.
But others, such as state Sen. Todd Weiler, R-Woods Cross, said that Republicans lowered their threshold from 70 percent to 60 percent in 1999 and since then voter participation has plummeted.
Former U.S. Rep. Merrill Cook said polls show public support for an alternative to the caucus system and Republicans who want to keep it need to realize it is in jeopardy.
"Whether we like to admit it or not, we're in a negotiation" with the Count My Vote group, Cook said. "If we don't up the threshold, we will in fact be losing our convention system."
It is still possible that the threshold to clinch the nomination could still be raised, but it would require a supermajority of all of the delegates at the Republican State Convention next month.
LaVarr Webb, one of the leaders of the Count My Vote movement, said the central committee's action makes it more likely his group will go ahead with the ballot effort
The initiative language is nearly drafted. There are deadlines coming up the group has to meet in order to have time to gather more than 100,000 signatures to put the initiative on the 2014 ballot.
"It's likely we're going forward, but the party still has one opportunity to make changes, so we'll see what happens there," Webb said.
Central committee member Arturo Morales said he would be a "foot soldier" in the fight to keep the caucus and convention system as it is and said that he and others opposing changes now need to make sure delegates don't support the change at the state convention.
"We have a lot of grass-roots groups across the state who are ready to do whatever has to be done to protect the caucus system," he said.
"It has been portrayed as we don't involve as many people, that these decisions within the Republican Party are just a select few, but in reality this select group that wants to have a say in the process are the ones funded by special interests," Morales said. "They want to keep power. They don't really care about the average voter."
Proponents argue the caucus system gives underfunded, unknown candidates a shot at knocking off incumbents. But it also has led to the defeat of popular incumbent candidates like Gov. Olene Walker and U.S. Sen. Bob Bennett.
Utah Republican Party Chairman Thomas Wright who had previously said he believed the ballot initiative would pass if the party didn't adapt said he was proud that he tried to broker a deal in his party and he can't predict where the issue goes now.
"I was hopeful today that we would pass proposals that would meet the objectives that were put forth by Republicans who have concerns about our caucus and convention system," said Wright. "They clearly stated what those concerns were, and this body did not agree to move in that direction."
The Count My Vote Group has floated the idea of letting a candidate qualify for a primary if he or she gathers signatures from 2 percent of the registered voters in the jurisdiction, rather than having to win support from the party delegates.
Democratic Party Chairman Jim Dabakis has said the clash over the nominating process is a Republican civil war, with old-guard Republican moderates like Leavitt "trying to steal back power," and said Democrats should stay out of the fight.
The Republican State Central Committee approved a proposal that gives the party more flexibility on when to hold neighborhood caucuses and how to cast the votes.
Wright said that change is significant and will make it easier to participate for those on religious missions, in the military or with work schedules or family commitments that conflict with the caucus meetings.
Committee members rejected proposals that would have opened the closed Republican primaries by allowing those who aren't registered members of the party to vote.
Republicans embrace status quo
The Republican State Central Committee rejected proposed changes, including raising the threshold for candidates to capture the party nomination outright at convention without a primary. Members defeated an effort to raise that threshold to 70 percent on a 56-74 vote and then rejected a move to require a two-thirds vote to secure the nomination on a 46-79 vote. Currently, a candidate needs the backing of 60 percent of convention delegates to win the nomination without a primary.
Republican leaders have one more chance to change the nominating rules at the state GOP Convention May 18.