The charismatic Cuban American lawmaker from Florida, the theory goes, could prompt Hispanics to consider supporting the GOP ticket — even after a primary contest in which dust-ups over illegal immigration have left some conservative Hispanics uneasy.

But Rubio’s role in recent controversies, including a dispute with the country’s biggest Spanish-language television network and new revelations that he had mischaracterized his family’s immigrant story, shows that any GOP bet on his national appeal could be risky.

Democrats had already questioned whether a Cuban American who has voiced conservative views on immigration and opposed the historic Supreme Court nomination of Sonia Sotomayor, the first Latina justice, could appeal to a national Hispanic electorate of which Cubans are just a tiny fraction but have special immigration status. And Rubio’s support in Florida among non-Cuban Hispanics has been far less pronounced than among his fellow Cubans.