We trusted free software for a long time before we needed to call it open source to make it sound palatable to less visionary business leaders.
We'll go on trusting the FOSS. The free and open nature of it means that we're never really trusting the company (the corporate entity) just the people they've amassed to work on the FOSS. Those people can continue to work on the FOSS after the company disolves, because the osftware is open/free.