The securities have a cusip and if the sale was done by a brokerage of repute - then the owneship would transfer
The buyer I assume gets ACCRUED INTEREST as is the case in a purchase of any bond
So as a seller you owe the accrued interest when you sell -- but the buyer should pay the agreed price which is a smidge higher before x date then when far from it
I buy premium and discount bonds all the time - and I have never seen an issue of the buyer or seller owing THAT non interest difference between PAR and the agreed to price
(interest paid inkind might be a wrinkle but right now its 1/4 of 13%)
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