Can FOSS adopt restrictions that favor OA?

Ben Brumfield has written software, FromThePage, to coordinate the work of online volunteers in transcribing and digitizing handwritten manuscripts.  Now he wants to release the code as open source, but only under a license that would require users to make the resulting transcriptions OA.  From the blog post on his dilemma:

...My quandary is this: none of the existing Free or Open Source licenses allow me to require that FromThePage be used in conformance with Open Access. Obviously, that's because adding such a restriction -- requiring users of FromThePage not to charge for people reading the documents hosted on or produced through the software -- violates the basic principles of Free Software and Open Source. So where do I find such a license? ...