I think Brythan's answer is technically incorrect. I will lay out some assumptions and explain my rationale.
I assume that the loan is not a HUD, VA, or USDA loan, and that it does not have mortgage insurance (80% loan to value) for simplification purposes. Along with the loan price (by far the most important factor), it conforms to GSA seller-servicer guidance and further falls under what is considered a "Qualified Mortgage". These assumptions are meant to conform to the bulk of mortgages sold in the US.
It is very doubtful that this loan has not been bought by the GSEs.
The private sector...controls $3.9 trillion in mortgages or 38.8% of the total loans outstanding. 
The vast number of the loans that the private sector originates are jumbos which this loan is not. Jumbos are defined as those above the lending limits of FHFA of $453,100. While possibly the GSEs did not purchase this loan, it's really not probable. In my estimation less than a 1% chance, and those are usually situations where a banking executive or board member holds the note. There would be no compelling reason given the capital requirements that a bank would ever hold a GSA compliant loan on it's books.
So what happens when FNMA, FRE, or GNMA purchase a loan? They either hold it in portfolio (held for investment, HFI) or sell it (held for sale, HFS). In addition, for loans sold, the GSAs are the guarantor of those loans. (More about this later). However, both portfolios are at least in part financed through loan sales. And the largest purchaser of those loans (but not sole) is the U.S. Federal Reserve.
And how does the Federal Reserve get the money for those purchases? They sell Treasury Securities (or they can print money). As Brythan indicated, that would add to the debt, offset by the mortgage asset on the balance sheet of the Fed.
In addition, the GSAs is the loan guarantor backed by the full faith and credit of the US Treasury. This is not an academic exercise. The US taxpayer is fully on the hook for the performance of the loans the GSEs purchased if they are sold to the Fed or some other third party. While this debt remains an unknown quantifiable until such time as a default may arise, it is an obligation to the taxpayer.
While it's impossible to state whether or not from your example that what I have outlined is the case, the plurality if not the majority of the loans you have described would fit this scenario.