This brief is well done (Carter Phillips/Evan Caminker) but also important in that it represents the views of Chief Justices of states from across the political spectrum. Tom Wolf tweet thread explains the brief’s significance. From the Brief’s summary of the argument:

The U.S. Constitution provides each State with authority over “the structure of its government.” Gregory v. Ashcroft, 501 U.S. 452, 460 (1991). Both before and after the Framing of the U.S. Constitution, the States authorized judicial review under state charters; and, at the time of the Framing, that state practice was adopted in the U.S. Constitution, see Marbury v. Madison, 5 U.S. (1 Cranch) 137 (1803), and in numerous state constitutions. See Jeffrey S. Sutton, 51 Imperfect Solutions: States and the Making of American Constitutional Law 13 (Oxford Press 2018). Further, many state constitutions from the Founding era contained provisions regulating elections. This historical context strongly supports state court review of state election laws under state constitutions. And while the text of the Elections Clause requires that state legislatures prescribe the laws governing federal elections, it does not otherwise displace the States’ established authority to determine the final content of their election laws, including through normal judicial review for constitutionality.

This conclusion is confirmed by the rest of the Elections Clause: the Clause specifies that Congress can override state election laws governing federal elections, yet Congress’s enactments are presumed to remain subject to constitutional review. State election laws likewise remain subject to state court review under the state (and federal) constitutions. State judicial review does not derogate from the primacy of the state legislature’s role. The legislature enacts state election laws and often plays a significant role in shaping the state’s constitution. And this Court’s precedent has explicitly and implicitly authorized significant checks on legislative power to make election laws, including a gubernatorial veto (see Smiley v. Holm, 285 U.S. 355 (1932)), state judicial review (Rucho v. Common Cause, 139 S. Ct. 2484 (2019)), judicial remedial authority (Growe v. Emison, 507 U.S. 25 (1993)), and state plebiscites (Ariz. State Legislature v. Ariz. Indep. Redistricting Comm’n, 576 U.S. 787(2015))….

Like their federal counterparts, state courts approach judicial decision making using a set of established tools. While they may not always use precisely the same interpretive frameworks as do federal courts—e.g., they may use different resources in determining the legislators’ or Framers’ intent in drafting a law or constitutional provision—they nonetheless are engaged in judicial review, not legislative acts, when they determine the content and constitutionality of state laws. The Elections Clause does not eliminate this consequence of our federal system, and it does not authorize the federal courts to impose their approaches or outcomes on state courts’ interpretation of state laws and state constitutions. Any federal review therefore must be exceedingly deferential.

The Conference is equally focused on the need for clear guidance about any constraints imposed on state courts by federal judicial review under the Elections Clause. Absent a clear standard, state courts will be unsure whether to apply otherwise applicable state laws and constitutional provisions, a consequence damaging to state sovereignty and judicial independence.

This concern is not addressed by the suggestion that state court judicial review is prohibited by the Elections Clause if it involves state constitutional provisions that are deemed too general or that impose substantive rather than procedural requirements. Such formulations do not provide state judges (or federal judges reviewing their decisions) with sufficient clarity to determine which constitutional provisions may be enforced, and the uncertainty will lead to disruptive litigation as state courts attempt to discern in expedited election-related proceedings which provisions they must disregard. Moreover, state courts have construed and developed precedent under so-called general constitutional provisions for decades, just as federal courts have done. Nothing in the Elections Clause suggests that state judicial review—unlike federal judicial review—should be cabined in this manner.