Elliott Investment Management has increased its equity stake in Southwest Airlines Co. to 9.7%, nearing the amount needed to call a special shareholder meeting at which the activist hopes to replace most of the carrier’s board. 

The larger holding, up from about 8% previously, was disclosed in a regulatory filing on Monday, just after Elliott renewed its call for sweeping leadership and operational changes at Southwest in a letter to shareholders. The activist needs a 10% stake in order to call a special gathering for Southwest investors to vote on its director nominees.

The latest volleys advance Elliott’s campaign to overhaul the carrier, which the activist has criticized for refusing to adopt changes embraced by rivals over the last 15 years, such as offering a bare-bones economy fare, charging customers for checked luggage and assigned seats.

Elliott has called for the ouster of Southwest Chief Executive Officer Bob Jordan and Chairman Gary Kelly and has named 10 people it would nominate for the airline’s 15-member board of directors. The activist’s economic interest in Southwest, which includes financial instruments such as derivatives, remained at roughly 11%.

Elliott said it’s eager to engage with Southwest’s board and plans to meet with the carrier on Sept. 9. However “so long as the jobs of Mr. Jordan and Mr. Kelly remain sacrosanct,” the activist said in its letter on Monday, “it is preferable to give shareholders a direct say on the question of who should be leading Southwest.”

The carrier repeatedly has backed its current leadership team, with Jordan recently telling employees the airline has an “incredibly strong plan” for its future that will be detailed at an investor day in late September. The airline already has also announced plans to adopt assigned seating on flights and offer premium seats.

“We welcome the opportunity to discuss ideas that would drive sustained shareholder value as we work to reach a collaborative resolution,” Southwest said in a statement on Monday. It also confirmed the Sept. 9 meeting with Elliott.

Southwest’s shares were little changed at 11:43 a.m. in New York. The stock had declined 2.4% this year through Aug. 23.

The carrier has said it’s open to a “constructive dialogue” to consider ideas Elliott might have to improve the business. Jordan has also warned in a message to employees that “if it’s a fight they want, it’s a fight they will get.” Jordan has said he has no plans to step down.

The airline last month named a new director who’s served as an executive at several airlines and adopted a poison pill shareholder rights plan to help fend off Elliott’s actions.