Matheson’s ad suggested voters shouldn’t be asked to weigh in on any changes to the park unless they are offered specific plans, rather than the more conceptual ones put forth by the tournament.

“The county is doing this backwards,” Matheson said. “The county commission should present to the public a firm plan that the public could look at and examine.”

On the contrary, Stearns said. If voters set the “outer limit” of what could be built, the county can later negotiate more favorable terms, including a shorter lease or requiring more money from the tournament.

“You can’t have a deal that involves facilities bigger or larger or grander” than what’s in the outlined plans, Stearns said.

Tournament organizers received the backing of Key Biscayne Village Council members, but only after the board stipulated certain conditions, including that the improved tennis center not be used to hold more tournaments or other events, such as concerts. The park is open to the public year-round except for the two weeks to play plus the time to set up and clear out.

“Our community probably, on balance, supports the Sony,” said Key Biscayne Mayor Frank Caplan, who is on the ballot measure’s steering committee. “We’ve become accustomed to the traffic impacts during the tournament itself, but it’s burdensome, and we don’t want those type of impacts to metastasize during the rest of the year.”

Caplan is working on getting the village, the tournament and the county, through district Commissioner Xavier Suarez, to agree to the stipulations in a written memorandum before Election Day.

Mayor Gimenez, for his part, told The Herald that he plans to vote for the measure. On Tuesday, his office issued a news release “encouraging” voters to vote on the measure, buried in a long ballot.

“It’s a county facility that needs to be upgraded, and any time that a private entity comes forward and says they will do it without using county funds, I think that’s a good thing,” he said, acknowledging that “devils are always in the details.”

“We’re not going to be putting forward anything that puts the county general fund in jeopardy,” he said.