Boeing Co. is working through the final steps before asking U.S. regulators to review an update for anti-stall software linked to two fatal 737 Max accidents, an early milestone to lifting a global grounding of its best-selling jet.

But there are many steps in the Federal Aviation Administration’s assessment of the proposed fix, and that process could stretch well into June even if there are no complications, said a person familiar with the matter. Boeing must also convince authorities from Beijing to Brussels that the plane is safe. Canada has already signaled it won’t follow an FAA panel’s recommendation against requiring additional simulator training for pilots.

Boeing has completed its engineering trial of the updated software, and its technical and engineering leaders were on board the final flight test earlier this week, Chief Executive Officer Dennis Muilenburg said in a video message late Wednesday. Up next is what he described as a “certification flight,” as Boeing prepares to submit the final paperwork to U.S. regulators.

For that flight, Boeing will hand over the controls of a 737 Max to FAA pilots to test design enhancements that the company says ensure the system won’t ever again overwhelm flight crews—as it did in two crashes that killed a total of 346 people. The regulator will determine when the certification flight takes place.

“We’re making steady progress toward certification,” Muilenburg said, standing in front of a Max aircraft at Boeing Field, an airport south of Seattle. Earlier in the day, he had been a passenger on a demonstration flight, watching the final update to the so-called MCAS software “operating as designed across a range of flight conditions.”

Muilenburg has stepped up Boeing’s campaign to boost public confidence in the safety of the 737 Max, and the company’s airplane designs, after two of the jets crashed within five months. The Max, which debuted in May 2017, is the newest version of a single-aisle jetliner family that is Boeing’s biggest source of profit.

In all, Chicago-based Boeing has conducted 120 flights, spending 203 hours in the air testing the new system, Muilenburg said. The campaign has included a 737 Max 7 outfitted with flight-testing instrumentation, as well as aircraft that have rolled out of a Boeing factory south of Seattle with the updated software already installed.

The upgrade is designed to make the anti-stall system less aggressive and prevent the repeated nose-down commands that overwhelmed flight crews for Lion Air and Ethiopian Airlines. In addition, MCAS—it stands for Maneuvering Characteristics Augmentation System—would no longer be triggered by a single erroneous sensor reading.

Southwest Airlines Co., the biggest U.S. operator of the Max, expressed confidence in the plane. Chief Executive Officer Gary Kelly said the carrier is eager to resume flying the aircraft once it’s ready. He also said Southwest had no plans at this time to cancel any Max orders.

“It’s a very good airplane but Boeing has acknowledged they have some things they need to address,” Kelly said at an event in Dallas on Thursday. “It seems like a relatively straightforward modification. We’re anxious to get the airplane back.