Boeing Co. is close to a significant milestone that’s one of the last steps before it submits to U.S. regulators a patch for anti-stall software linked to two fatal 737 Max accidents.

The company 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 the certification flight, he said, as Boeing prepares to submit the final paperwork to the Federal Aviation Administration.

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 wouldn’t ever again overwhelm flight crews, as it did in two crashes that killed 346 people. The regulators will determine when the certification flight takes place.

“We’re making steady progress toward certification,” Muilenburg said, with a Max aircraft and Boeing Field, an airport south of Seattle, as backdrops. 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 been stepping 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.

Boeing’s ultimate goal is “to make the 737 Max one of the safest airplanes to ever fly,” Muilenburg said.