Is expiation obligatory with doubt regarding the expiration of the oath's term, given that the swearer predominantly believes the term has expired?

If you believe it more likely that you swore an oath on the first day of the month, then act accordingly. If ten days pass from that date and you do the thing you swore an oath about, then you have not broken your oath; because strong conjecture is treated as certainty, and it is upon this that religious rulings are built.