Two of the nation's largest airlines on Tuesday provided updates that suggest the worst of the COVID-19 induced travel slump might be behind us, encouraging signs for an industry feared to be headed for a liquidity crisis if travel demand does not quickly return.

Southwest Airlines (NYSE:LUV) in a regulatory filing said it "has recently experienced a modest improvement in passenger demand, bookings, and trip cancellations, resulting in month-to-date net positive bookings ... where new passenger bookings outpaced trip cancellations."

Net bookings were negative through most of March and April. Southwest said it expects its load factor, roughly a measure of how full its planes are, to be in the range of 25% to 30% for May, an improvement over the previous estimate of 5% to 10%.

A plane landing at night.

Image source: Getty Images.

Southwest said it "has also recently experienced a modest improvement in passenger demand and bookings in June 2020."

United Airlines Holdings (NASDAQ:UAL) in a separate filing said that while gross bookings were down 95% year over year in April, it is seeing "a reduction in customer cancellation rates and a moderate improvement in demand" for travel.

These numbers do not imply the crisis is over. Southwest still expects operating revenue to be down 80% to 85% in June and United for June has chopped 90% of its June 2019 schedule. It's going to take months, if not longer, for schedules to normalize, and could take three years for traffic to return to pre-pandemic levels.

But they do suggest demand might have bottomed out last month and is slowly starting to recover. Airline shares have lost 50% or more of their value so far this year due to the crisis, and investors remain concerned that if traffic does not return by Fall some of the carriers will run into liquidity issues.

If these trends continue to creep higher, and are experienced throughout the industry, that's one small but positive signal suggesting the industry can make it through the COVID-19 crisis without bankruptcies.