Instead of a little time from a lot of employees (which might be as much a burden for the project as a blessing), it might be better for many companies to pay into a pot that provides someone's salary, while one of the companies donates the desk. This is what I meant by administration.

An issue with that is that even if you use a FOSS solution, your devs might not have the domain experience to contribute. Of if they could, it may be pretty inefficient to or take a while to get up to speed to get any meaningful contributions in.