I think this is the truth but it is not verifiable like you said.
He was a visiting lecturer not an adjunct and had no qualifications to teach Constitutional Law from all accounts. Personally would think Tom Ayers is the one who talked to the Board to get him on the payroll and give him an office. He was tight with his son Bill Ayers and by all accounts Tom Ayers was one of the ones behind Obama getting into Harvard. Since Tom Ayers sat on the Board of Trustees at Northwestern and his son was a professor at University of Chicago I can see that happening.