Three men who say Mark Zuckerberg stole the Facebook concept from them have asked a US appeals court to re-open a $65m (£42m) legal settlement signed in 2008 with the company.
Cameron and Tyler Winklevoss and Divya Narendra argue Facebook undervalued its share price when they struck the deal, depriving them of millions of dollars.
Facebook maintain that the claims for more money are without merit.
A decision on whether the case will be heard again is expected before April.
A lower court has already ruled the earlier accord was binding and should stand.
But if the Ninth Circuit Court of Appeals in San Francisco gives the go ahead for the men's request to sue Facebook all over again, it would mean they can pursue Facebook for more money.
They will also hope to extract an admission that the idea for the site was stolen when Mr Zuckerberg did some work for them during their time at Harvard University.
Facebook agreed to the 2008 settlement to end "rancorous litigation" but did not admit Mr Zuckerberg had taken their idea.
Mr Zuckerberg, who was at Harvard with the three men, has always said Facebook was his creation.
Today it is the world's biggest social network with over 500m users, and Mr Zuckerberg is one of the world's youngest billionaires.
This long-running legal battle, which was immortalised by Hollywood in a hit movie last year, played out in an altogether different venue for the sequel.
The courtroom was told the $65m deal included $20m in cash and $45m in shares. Those shares are now worth roughly $140m.
The valuation used was one that put Facebook's worth at $15bn, based on an investment that had been made by software giant Microsoft.
Lawyers for the Winklevoss twins and Mr Narendra said it was discovered several months after signing the deal that Facebook had been internally valued at under $4bn.
Lead attorney Jerome Falk told the court that had that lower price been used as the basis for his clients' settlement, they would have been awarded four times the 1.25m shares they got.
In today's prices, that would add up to about $600m in share value.
Mr Falk said an accurate share price was not disclosed to the founders until after they signed the term sheet.
He argued that this information was material to the settlement and that Facebook had a duty to tell his clients about it.
Facebook's lawyers disagreed, saying the internal valuation was just one opinion and that the Winklevosses had had ample conflicting valuations of the company's shares before the mediation started.
They were not interested in pinning down a monetary value, said Joshua Rosenkranz, Facebook's lead attorney. "They were interested in owning a specified portion of the world's hottest start up," he said.
For about an hour the three-judge panel heard arguments from both sides. They peppered the legal teams with a variety of questions ranging from the merits of mediation to points of law.
Senior Judge Clifford Wallace said that the twins had several lawyers representing them at the earlier settlement talks, and that that their father was a business expert, factors he suggested made it hard to believe that anyone took advantage of them.
"I agree my clients were not behind the barn door when brains were passed out," said Mr Falk, referring to the Winklevoss twins.
At the end of his arguments, Mr Rosenkranz said the three men had struck a deal that had made them rich and was making them richer by the day, and suggested it was time to move on.
Throughout the proceedings the Winklevoss twins, who are elite rowers and took a break from their training for the 2012 Olympics to attend, sat in the front row and remained stony-faced.
After the hearing they declined to comment except to say that they looked forward to the court's decision.
It was a sentiment echoed by Facebook.
"We appreciate the court's time and look forward to their ruling," said company spokesman Barry Schnitt.
A judgement is expected in two to three months.