Odds still favor the eventual construction of Barack Obama’s $500 million presidential museum and library in a public park along Chicago’s lakeshore. But it’s no longer a sure thing in the face of a formidable legal challenge by a parks-advocacy group.

A federal judge in Chicago hears arguments Thursday on a city motion to toss the group’s lawsuit that argues the project runs afoul of laws barring development in parks hugging Lake Michigan.

A ruling for Protect Our Parks could signal the Obama Presidential Center is in real trouble.

A look at the project and some of the key legal issues:

Q: WHAT ARE THE PLANS?

A: If backers of the center prevail, the center would be built 7 miles (11 kilometers) south of downtown Chicago in Jackson Park, named after President Andrew Jackson. The park, landscaped in the 1800s and a site for the Chicago World’s Fair in 1893, is listed on the National Register of Historic Places. It’s near low-income neighborhoods where Obama once worked as a community organizer. It’s also just blocks from the University of Chicago, where Obama was a law professor and near where the Obamas lived until he won the presidency in 2008.

The center would comprise 20 acres (8 hectares) of the 500-acre (202-hectare) park. Its centerpiece building would be a 225-foot (69-meter) museum tower, with a cluster of lower buildings around it, including a 300-seat auditorium. The center’s website says the complex will be “a world-class museum and public gathering space that celebrates our nation’s first African American President and First Lady (Michelle Obama).”

It was originally slated to open in 2021, though ground hasn’t yet broken.

Q: WHAT’S THE AGREEMENT WITH CHICAGO?

A: The Obama Foundation, a private nonprofit, would pay the costs of constructing the complex. It would also be responsible for covering operating costs over the length of the 99-year agreement. Once built, however, the Obama Presidential Center’s physical structures would be transferred to the city for free. So the city would formally own the center but not control what happens at it.

Among the steps taken to make the park land available for the project, the Chicago Park District first sold the land to the city for $1. Illinois legislators — at the city of Chicago’s urging — also amended the state’s Illinois Aquarium and Museum Act to include presidential libraries as an exception to the no-development rules if there’s a compelling public interest. The Chicago City Council approved the project by a 47-to-1 vote last May.

Q: WHAT’S THE CORE OBJECTION?

A: In its 2018 suit , which only names the city and park district as defendants, Protect Our Parks accused the city of illegally transferring park land to a private entity, The Obama Foundation. They say city officials manipulated the approval process and tinkered with legislation to skirt long-standing laws designed to ensure residents had unobstructed access to lakeside parks.

“Defendants have chosen to deal with it in a classic Chicago political way … a short con shell game, a corrupt scheme to deceive and seemingly legitimize an illegal land grab,” the lawsuit says. It also described the city as “gifting” prized land to a Chicago favorite son.

Democrat Obama remains hugely popular in overwhelmingly Democratic Chicago and most Chicagoans back the project.

But in a friend-of-the-court brief supporting Protect Our Parks, legal scholar Richard Epstein said public-trust doctrine places an extra burden on authorities to prove overwhelming public benefit when they offer the use of public parks to such popular, well-connected figures as Obama.

“Obama is one of the most powerful and influential personages in Chicago life, with deep ties to Mayor Rahm Emanuel (Obama’s former White House chief of staff) and with many close connections to key city public officials,” Epstein wrote. “His enormous clout cries out, not for deference, but for searching scrutiny” of whether benefits of the project far outweigh the loss of public-park access.

Among other assertions in the lawsuit is that the center would interfere with migrating butterflies and birds.

Q: THE CITY’S DEFENSE?

A: City lawyers say Protect Our Parks misreads the law , misrepresents how the approval process played out and exaggerates potential environmental disruptions. They say planners have done all they can to blend center facilities into the surrounding nature, including by planting more than 400 new trees.

City lawyers say they have demonstrated significant benefits, including that the center would provide a major economic boost to economically hard-hit minority communities in the vicinity. Backers estimate it’ll create 5,000 jobs during construction and over 2,500 permanent jobs. An estimated 760,000 people could visit each year, foundation officials say.

Thirteen presidential foundations, including the George W. Bush Foundation and Ronald Reagan Presidential Foundation, also threw support behind the project, saying in a joint filing that “it is neither unusual nor improper for the Obama Presidential Center to be linked to a non-governmental foundation.”

And a friend-of-the-court brief submitted by 11 existing Chicago museums said the center in Jackson Park would “be a cultural and economic treasure for Chicago.”