PkmnQ is cool
The only remaining unconfirmed-position prime left is M136279841. It could technically be anywhere from the 52nd to the 266697th, there's no way to know until the double-checking's all done.
According to the "Est Complete" section of the table, it should complete today. Either way, it's on its last two percent, so whatever the case, M82589933's status as either the 51st or 52nd Mersenne prime (but the latter case is unlikely).
Kriesel has finished M81648221, the final exponent to be tested is M81821569, which will be done by Greg
ooh apparently we're at the last two un-double-checked exponents below M82589933, and they'll be done in a few days, so we'll officially know if M82589933 is the 51st Mersenne prime or if there's secretly another one just below it soon