NEW DELHI (Reuters) -Airline bosses are tentatively voicing greater confidence in Boeing's ( BA ) ability to deliver jets at the right quality in a step towards ending years of reputational damage for the embattled planemaker.

An annual summit of airline leaders in New Delhi this week struck a more optimistic tone about Boeing's ( BA ) recovery from overlapping safety, regulatory and industrial crises, though executives stressed Boeing ( BA ) still had much work left to do.

"What we've seen quarter to quarter is an improvement in safety, an improvement in quality," Seattle-based Alaska Airlines CEO Ben Minicucci told Reuters at the International Air Transport Association's annual meeting.

"Are they yet there? No, there's still a lot of work."

In January last year, a door missing four bolts blew off a new Alaska Airlines Boeing 737 MAX jet at 16,000 feet. The ripple effect from the incident caused a crisis throughout the aviation industry, hitting suppliers, carriers and passengers.

Planemakers often have to run the gauntlet of airline criticism at IATA's annual meeting, where IATA head Willie Walsh last year urged Boeing ( BA ) to learn from errors that fuelled a crisis beginning with fatal crashes of two MAX jets in 2018 and 2019.

One of Boeing's ( BA ) most vocal recent critics has been Tim Clark, president of Emirates, the largest customer for big jets like the Boeing 777 and its long-delayed successor the 777X.

At his annual sit-down with reporters at the summit this year, Clark spoke more positively about getting "clearer messages" from Boeing's ( BA ) recently appointed leadership.

He welcomed a change of style under CEO Kelly Ortberg, revealing he had never met Ortberg's ousted predecessor Dave Calhoun.

"It was nice to meet the head of Boeing ( BA )," he said.

Clark, whose airline has 205 of the still uncertified 777X on order, suggested growing confidence from Boeing ( BA ) in private.

"When I talk about cautious optimism, in the last few years I had seen none of that" he said.

Ortberg, who took the helm last August, has said he will address safety and quality concerns and repair trust with regulators, staff and customers.

Boeing ( BA ) customers said the measures are showing some results.

While Boeing ( BA ) is still trying to ramp up production, there have been improvements in quality and visibility of deliveries.

"There is still further to go but it is definitely an improving story," said Peter Barrett, CEO of lessor SMBC Aviation, a major leasing company and Boeing ( BA ) customer.

AIRLINE INSPECTORS

Few are taking Boeing's ( BA ) recovery solely on trust.

Alaska Airlines has sent its own quality inspectors to Boeing's ( BA ) production lines and commissioned a quarterly audit.

Alaska's Minicucci said Boeing ( BA ) had put in a "very diligent, deliberate process" to reduce out-of-sequence work, a disruption to normal factory flows blamed in part for the oversight of the missing bolts.

He noted that Ortberg, who has moved to Seattle to be closer to the biggest jet plants, and Boeing Commercial Airplanes CEO Stephanie Pope, appointed last March, regularly tour factories.

"They're getting out there, they're walking the floor, they're feeling what's going on," Minicucci said. "I think that's different to what happened in the past."

Other customers have noted progress, despite a bruising strike last year which halted most of Boeing's ( BA ) jet production.

United Airlines CEO Scott Kirby, who took part in a revolt by U.S. airlines that triggered Calhoun's exit after the Alaska incident, said last week Boeing ( BA ) had "turned the corner".

WORK REMAINS TO COMPLETE RECOVERY

Still, few airline chiefs have any illusions about the scale of turnaround needed to restore the status of Boeing ( BA ) factories.

"When I was there last year... I thought this is going nowhere at this rate," Clark said, adding he would return this year to see for himself the progress made since then.

Boeing ( BA ) declined to comment on specific airline opinions.

Nor does the brighter public tone mean airlines will necessarily ease pressure in private as they wait for new jets.

But several airline executives at the IATA event acknowledged that Boeing ( BA ) had passed one key test of support at the industry's biggest annual podium, as it slowly stabilises production.

"We have got a lot of work still to do (but) there is a lot of positivity from customers on our performance in the first five months of the year," Boeing ( BA ) Senior Vice President of Commercial Sales Brad McMullen said.

It now faces a decisive test as regulators review higher output after Boeing ( BA ) hit a temporary ceiling of 38 MAX jets a month. It also seeks certification of models including the 777X.

The acting head of the Federal Aviation Administration said on Wednesday higher output would not happen straightaway.

Originally due to receive the 777X in 2020, Clark held out little hope of getting the upgraded jet before IATA meets again in June next year. Boeing ( BA ) has said it is now due in 2026.

"The important thing is they get it out and it's certified to the rigours of the new Boeing ( BA ): the new approach to building aircraft, safety of operation and all the quality controls that they were having difficulties with prior to that," Clark said.