Market Basket CEO Arthur T. Demoulas and the Massachusetts-based company's board of directors agree on one thing after publicly feuding for months: They're both hoping mediation talks that start today lead to a "successful" resolution of the supermarket chain's internal fight.

The closed-door talks are happening in Delaware, where the company is incorporated. Demoulas, who was suspended by the board in May, will take part in the proceedings. 

"Mr. Demoulas firmly believes that mediation represents the best path to resolution of the issues facing Market Basket and is committed to reaching agreement," Demoulas spokesperson Justine Griffin said in a statement. "He is confident that if the board shares that commitment, mediation will be successful."

Harvey Wolkoff, an attorney for the board of directors, told WBZ-TV he could not reveal how the board is approaching the mediation process.

"However, we are very hopeful that at the end of the mediation, which is scheduled for one, or, if necessary, two days, that we are able to report a mutually successful resolution," Wolkoff said.

Market Basket latest developments

The new Market Basket fight is happening 11 years after Arthur T. Demoulas was first ousted as CEO by his cousin, Arthur S. Demoulas. An unprecedented customer boycott ensued and employees loyal to "Artie T" walked off the job. He was able to get his job back and buy the company along with help from his three sisters.

But the board that's controlled by his sisters suspended Demoulas this year, saying that he was being investigated for "planning a work stoppage" and not giving the board key financial information, or a succession plan. He called the suspension an excuse for a "hostile takeover."

Since then, two executives loyal to Demoulas have been fired, while some employees claim the corporate culture at Market Basket is now one of fear and hostility. 

Market Basket mediation

Boston Business Journal reporter Grant Welker told WBZ-TV that Demoulas may need to allow the board to have more say in Market Basket's operations in order to keep his job. It also remains to be seen if Demoulas will be allowed to bring back the fired executives and reinstate his children in their company positions.

"A lot of those things need to be buttoned up if they are to say 'We want to keep him on, but what does that look like moving forward?'" Welker said.

This mediation session could be a "turning point" for Market Basket's future, Welker believes.

"A lot of family business experts say that it's increasingly difficult with each successive generation to keep the company going as it was, as being family owned and operated," he said. "There could be a lot of questions about what would a post-Arthur T. Demoulas look like?"

Shoppers concerned

As the bitter fight plays out behind the scenes, shoppers at the Burlington location said their loyalty is being tested.

"I'm personally afraid that Market Basket becomes like every other supermarket. Expensive, hard to navigate," said David Vaudo, who said he got his first job at Market Basket when he was 16.

Boston Business Journal's Grant Welker said even if Demoulas is reinstated, the future of Market Basket remains uncertain.

"This environment that people love working in and where people love shopping at, it's at least up in the air," said Welker. "Is he able to bring back those executives who were fired in July? Would the board have a say on that?"