You can find the majority opinion in Murthy v. Missouri of Justice Barrett, along with the dissent of Justice Alito (joined by Justices Gorsuch and Thomas) at this link.

The claim was that government agencies pressured or coerced social media platforms including Facebook and Twitter to remove content (related to the election, Covid, etc.). This what the term “jawboning” refers to.

The Court did not opine on what would have to be proven in a jawboning case involving social media companies, because it held that none of the plaintiffs had standing: they did not show enough of a connection between the government‘s actions and plaintiffs’ injuries. As the majority opinion states: “the platforms moderated similar content long before any of the Government defendants engaged in the challenged conduct. In fact, the platforms, acting independently, had strengthened their pre-existing content moderation policies before the Government defendants got involved.”

Given that the majority said it would not reach the merits of the jawboning question, it’s inclusion of footnote 4, casting aspersions on the ridiculous factfinding of the district court, was notable as a slam. This is arguably the most important part of the opinion:

The Fifth Circuit relied on the District Court’s factual findings, many of which unfortunately appear to be clearly erroneous. The District Court found that the defendants and the platforms had an “efficient report-and-censor relationship.” Missouri v. Biden, 680 F. Supp. 3d 630, 715 (WD La. 2023). But much of its evidence is inapposite. For instance, the court says that Twitter set up a “streamlined process for censorship requests” after the White House “bombarded” it with such requests. Ibid., n. 662 (internal quotation marks omitted). The record it cites says nothing about “censorship requests.” See App. 639–642. Rather, in response to a White House official asking Twitter to remove an impersonation account of President Biden’s granddaughter, Twitter told the official about a portal that he could use to flag similar issues. Ibid. This has nothing to do with COVID–19 misinformation. The court also found that “[a] drastic increase in censorship . . . directly coincided with Defendants’ public calls for censorship and private demands for censorship.” 680 F. Supp. 3d, at 715. As to the “calls for censorship,” the court’s
proof included statements from Members of Congress, who are not parties to this suit. Ibid., and n. 658. Some of the evidence of the “increase in censorship” reveals that Facebook worked with the CDC to update its list of removable false claims, but these examples do not suggest that the agency “demand[ed]” that it do so. Ibid. Finally, the court, echoing the plaintiffs’ proposed statement of facts, erroneously stated that Facebook agreed to censor content that did not violate its policies. Id., at 714, n. 655. Instead, on several occasions, Facebook explained that certain content did not qualify for removal under its policies but did qualify for
other forms of moderation.

Justice Alito, in contrast, found enough evidence of jawboning to find standing (and then a likely violation of the law by the government). He relied in part on a report from Jim Jordan’s “weaponization of government” committee in the House, something that itself is quite unreliable.