The brief addresses the question posed by the Court about whether the merits panel is bound by the motions panel.  Here is the summary of the argument:

Materially identical issues arise in deciding whether to maintain a preliminary injunction during an appeal of that injunction and during the rest of the case. The stay panel fully addressed these issues. Its decision is binding circuit precedent because it is published, and it is law of the case because the Court considered extensive briefing and oral argument.

Texas argues that the stay factors mirror the issues on appeal.

The stay factors therefore mirror the issues in this appeal: “The factors to be considered in deciding whether to stay an order pending appeal are virtually the same as the factors used by a court in deciding whether to issue a preliminary injunction.” Voting for Am., Inc. v. Andrade, 488 F. App’x 890, 893 (5th Cir. 2012); see Veasey v. Perry, 769 F.3d 890, 893 n.1 (5th Cir. 2014). Other circuits agree that “[t]he standard for granting a stay pending appeal mirrors that for granting a preliminary injunction.” In re A&F Enterprises, Inc. II, 742 F.3d 763, 766 (7th Cir. 2014); accord, e.g., Ohio Republican Party v. Brunner, 543 F.3d 357, 361 (6th Cir. 2008).

There is a slight temporal difference: A stay pending appeal lifts the injunction during the appeal, whereas a reversal lifts the injunction for the rest of the case. This difference does not necessarily entail different legal or factual arguments, however, and Defendants did not make any. In both their moving papers and briefing, Defendants deployed a full arsenal of argumentation.

…

In short, Defendants’ stay motion presented this Court with the occasion to decide every issue necessary to resolve this appeal, and the Court issued a lengthy published opinion doing so. The Court’s resolution of these issues is correct, as explained below in Part II. But there is no need to reconsider these rulings, as the Court’s decision sets circuit precedent and law of the case.

Texas further argues that the published opinion of the motion panel has “binding force.”

Published opinions by a three-judge panel set binding law of the circuit: “Published panel opinions are ordinarily binding on subsequent panels.” Camacho v. Tex. Workforce Comm’n, 445 F.3d 407, 411 (5th Cir. 2006); see, e.g., Samaad v. City of Dallas, 922 F.2d 216, 219 & n.2 (5th Cir. 1991) (collecting cases); United States v. Short, 181 F.3d 620, 624 (5th Cir. 1999) (noting that a “panel is bound by the precedent of previous panels absent an intervening Supreme Court case explicitly or implicitly overruling that prior precedent.”). If a party or a panel believes that a published opinion by a previous three-judge panel is erroneous, then the “remedy [is] to seek en banc consideration rather than to ask one panel of the court to set aside the ruling of another panel.” McClain v. Seaboard Coast Line R.R. Co., 490 F.2d 863, 864 n.3 (5th Cir. 1974) (per curiam); see 28 U.S.C. § 46.

The stay panel here published its opinion, and properly so.

The 5th Circuit had cited Mattern v. Eastman Kodak as a possible grounds that the motions panel does not bind the merits panel. Texas rejects that precedent, as it involved an unpublished decision.

This Court has noted that a merits panel has authority to overturn a motions panel’s ruling where the motions panel had not published a precedential decision. See Northshore Dev., Inc. v. Lee, 835 F.2d 580, 583 (5th Cir. 1988) (“the motions panel did not assign any reasons for its decision in an opinion”); Fischer v. DOJ, 759 F.2d 461, 463 (5th Cir. 1985) (per curiam) (motions panel “did not file an opinion assigning reasons for its decision”); see, e.g., Mattern v. Eastman Kodak Co., 104 F.3d 702, 704 (5th Cir. 1997); EEOC v. Neches Butane Prods. Co., 704 F.2d 144, 147 (5th Cir. 1983); United States v. Bear Marine Servs., 696 F.2d 1117, 1119 (5th Cir. 1983).

But the 5th Circuit has never held that published decisions could be viewed as non-precedential.

But this Court has never held that merits panels can choose to treat published decisions of motions panels as non-precedential. Any such rule would be an unnecessary drain on judicial resources. When a three-judge panel expends significant time and effort considering certain legal issues—by taking extensive briefing, holding lengthy oral argument, and publishing a thorough opinion—that decision is treated as precedent to maintain the consistency of circuit law and conserve the resources of future panels.

Of course, most motions-panel decisions will not warrant extensive consideration, and those decisions will remain unpublished—if reasons are given at all. But when a motions panel is called upon to analyze and resolve weighty legal issues of national importance, circuit rules—and common sense—dictate that the panel will publish a precedential ruling to prevent unnecessary duplication of effort. The stay panel’s published opinion here sets circuit precedent and binds future three-judge panels.

Further, under the “discretioanry” law of the case doctrine, the merits panel’s extensive consideration binds the merits panel:

The stay panel’s ruling here sets law of the case by any conceivable measure. Unlike many motions panels, the panel here “ha[d] the benefit of full briefs and . . . oral argument.” Bear Marine, 696 F.2d at 1119, cited in Mattern, 104 F.3d at 704. Cf. Whole Woman’s Health v. Cole, __ F.3d __, 2015 WL 3604750, at *12 (5th Cir. June 9, 2015) (per curiam) (merits panel not bound by motions panel that only had “an abbreviated proceeding”);

…

Due to the expedited briefing schedule sought by Defendants in this appeal, the stay panel was able to consider not only the motions briefing and full factual record, but also the merits briefing, a court-ordered supplemental brief preceding oral argument, over two hours of oral argument, and all of the amicus briefing. See, e.g., Op.*1 nn.9, 10, 12 (discussing merits briefing); Op.*6 n.40 (noting two-hour-plus oral argument). The thoroughness of that review is borne out in the extensively footnoted 42-page slip opinion and the corresponding 26-page dissent.

The rest of the brief addresses the merits.