Seems the likeliest scenario to me.
But I don’t think his foreign born scam could have been enough; there had to be somebody behind his application. Some promise was made to get him in, made by somebody big with deep pockets.
Such a transaction may have left a trail.
It’s pretty well-known that Khalid Almansour (adviser to Saudi Prince Alwaleed of Fox News fame) greased the skids for young Obama at Columbia.
Likely Obama got credit for academic work he never did in exchange for Saudi money.