There's not an explicit rule of professional responsibility for this, but the general commandment to serve the public ought to stretch far enough to keep the Court from running a propaganda arm. Amy Coney Barrett blasted a sound byte reassuring the public that the Court isn't a corrupt, partisan institution because it issues "opinions," knowing full well that she was about to obliterate half a century of environmental regulations on the strength of an unsigned paragraph that very week. We also have to revisit the Yale Law School culture fight, which seems less a fight than a unilateral assault on free speech in favor of "you have the right to shut up and listen."
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Above the Law - Thinking Like a Lawyer