9/11 Families United, which has previously written to agents of several American LIV players to express their outrage at them competing in the Saudi-funded events, has pledged to protest at Augusta National if the policy is not reversed.
The PGA Tour has suspended members who have competed in the breakaway competition without permission, while the DP World Tour fined players £100 000 and banned them from the Genesis Scottish Open, but saw that temporarily stayed on appeal.
R&A chief executive Martin Slumbers said in July that banning LIV players from next year's Open was "not on the agenda", but did not rule out changing the championship's entry criteria.
Masters chairman Fred Ridley took a similar stance on Tuesday, saying in a statement that the entry criteria for 2023 would remain the same, with any possible future changes announced in April.
Former Masters champions Phil Mickelson, Dustin Johnson, Sergio Garcia and Bubba Watson are among the top players to have joined LIV, while others who remain inside the world's top 50 - despite LIV not currently being allowed to award ranking points - will still be eligible to compete at Augusta National.
Ridley's statement drew an angry response from 9/11 Families United, who pointed out in their letter to the agents of Mickelson, Johnson, Reed, Bryson DeChambeau and Kevin Na in June that "Osama bin Laden and 15 of the 19 September 11 hijackers were Saudis."
Tuesday's statement from the organisation read: "In the aftermath of 9/11, our country agreed we would 'Never Forget' that horrible day.
"The only reason the Saudis launched LIV was to try to make the world forget who they are and what they did, including their role in 9/11.
"Anyone who truly vowed to 'never forget' should be appalled by the decision by these golfers to put money ahead of their own country.
"On behalf of 9/11 Families United, we are calling on Augusta National to reconsider their open-door policy to the LIV golfers. If they are welcomed with open arms, we will be at their front door to protest in April."