You can find the filing here.

I think that they have a strong argument about application of the Purcell Principle:

Applicants rely heavily on Purcell v. Gonzalez, 549 U.S. 1, 4-5 (2006), in which this Court cautioned that “[c]ourt orders affecting elections, …can themselves result in voter confusion and consequent incentive to remain away from the polls,” a risk that increases “[a]s an election draws closer.” The underlying purpose of this so-called “Purcell principle” is to avoid “changing the electoral status quo just before the election,” which would cause “voter confusion and electoral chaos.” Richard L. Hasen, Reining in the Purcell Principle, 43 Fla. St. U. L. Rev. 427, 428 (2016).

But in this case, the “electoral status quo” already has been upended―not by any judicial order, but by the COVID-19 pandemic and the “voter confusion and electoral chaos” it is causing. Until very recently, Wisconsin voters reasonably expected they would be able either to vote safely in person on election day or through a reliable, well-functioning absentee ballot system. Now they cannot―and not because of any court order, but because of the pandemic. See Op. at 39 (“election deadlines have already been disrupted, with the evidence showing that many voters who timely request an absentee ballot will be unable to receive, vote, and return their ballot before the receipt deadline”) (emphasis in original).

Likewise, this Court in Purcell was concerned that pre-election judicial orders might create a “consequent incentive to remain away from the polls.” 549 U.S. at 5. Here again, the “incentive to remain away from the polls” in these consolidated cases results not from federal judicial action, but from a deadly pandemic. Voter confusion and abstention from voting are “consequent” of COVID-19, not of anything the district court has done. That court’s limited relief will result in more voters being able to cast their ballots and ensure those ballots will be counted, rather than threatening to keep voters from casting their ballots. The court’s injunction, in other words, is an effort to mitigate the confusion and chaos caused by the pandemic that are interfering with voters’ reasonable expectations and threatening to keep them from voting. Indeed, it is telling that the WEC ― the body responsible for administering election ― is not challenging the district court’s order or arguing that extending the ballot receipt deadline to April 13 will cause disruption.

Applicants’’ reliance on Purcell is misplaced for yet another reason. It will be much less confusing and unfair if the district court’s order—which has received substantial publicity and is in the process of being implemented by the WEC and election clerks throughout Wisconsin4—simply remains in effect rather than being changed less than 48 hours before the election, which would lead to yet further confusion and uncertainty, as well as harm to voters who already have relied on the district court’s order in planning how and when to vote. Granting a stay, in other words, would simply create Purcell problems of its own, including by forcing voters who had planned to vote absentee but not yet received their ballots to either vote in person on election day and risk exposure to the coronavirus, or abstain from voting altogether.