I expect the district court, which just issued its order, will deny the request for a stay. This is more of a formality before seeking relief in the 11th Circuit.
This could end up before the Supreme Court in a matter of days.
As to what the courts will do, this depends upon how courts balance the equities (and liberal and conservative judges may do so differently). On the one hand, people are potentially disenfranchised without a chance to challenge disenfranchisement based upon an election official comparing two signatures (maybe from decades apart) to see if they match. On the other hand, this litigation is being brought late in the process, and the courts, under the Purcell Principle, have looked down upon late changes in voting rules by courts before the election.