For Diaz-Balart, the absence of the word communism means nothing. “Why is there a CAA? Is it because Cubans are nicer, more attractive, have better shoes? No. It’s because of the political conditions on the island,” he said. “And when you have people going back and forth, that’s an abuse.”

Rubio has said he wants to keep the CAA but so far made no specific proposals for reforming it. Ros-Lehtinen has only hinted that the law’s benefits should be denied to any Cubans in the United States who return to the island too quickly.

Diaz-Balart said he favors keeping the overall law and would not comment on any plans for legislative proposals on how to change it, apparently to keep Congressional opponents of changes in the dark until the last minute.

“The migration policy of the United States will not be decided by Cuban octogenarian dictators. We will be watching what will happen in these next few months, to act accordingly,” said Ros-Lehtinen.

Rep. Joe Garcia, a South Florida Democrat and also Cuban-American, said Democrats in Congress and the White House oppose changing the law because “the conditions that led to the Cuban Adjustment Act are the same that exist today” — a repressive government.

“I have spoken to the (immigration reform) negotiators in the House, in the Senate and in the White House, and the words Cuba or Cuban Adjustment Act did not come up in those meetings,” Garcia said.

There have been attempts through the years to change or abolish the act.

Former Rep. Barney Frank, a liberal Massachusetts Democrat, proposed getting rid of it in 2006. And former Rep. David Rivera, a South Florida Republican, in 2011 proposed denying its benefits to Cubans who return to the island for visits

Twelve leading Miami-based Cuban exile groups, including Bay of Pigs veterans in the Brigade 2506, issued a declaration Jan. 15 warning that unless the CAA is modified the United States will face a mass inflow of Cubans.

Havana’s migration reforms, designed to allow Cubans to travel abroad and return more easily, will unleash “a bothersome social political and economic avalanche,” the declaration said. The CAA, the groups declared, should benefit “only true political and ideological opponents, those repressed or persecuted for dissenting from the Communist tyranny.”

While the Cuban government has long condemned the CAA as an “assassin law” that lures Cubans into risky escape attempts aboard makeshift boats, Havana’s second-ranking migration official, Lamberto Fraga, noted last month that Havana’s decision to extend the time Cuban citizens can spend abroad without losing their island residency from 11 to 24 months meant a Cuban can take advantage of the adjustment act — “and retain his Cuba residency.”

Peters, who favors closer U.S. engagement with Havana, backs total abolition of the CAA “because it responds to a circumstance that no longer exists” and offers Cubans economic benefits, such as health insurance and child care, that can better be used for true refugees.