Various folks contend that they subscribe to some of the tenets of judicial supremacy, but not the full-fledged version that I describe. But the challenge for them is whether there is any principled basis for drawing the line where they do.
Genuinely “co-equal” interpretive authority and “judicial supremacy” (to any degree) are fundamentally inconsistent with one another. If the judgments of the judiciary are supreme and binding on everyone else, and if the judiciary is likewise supreme in deciding the scope and content of such judgments and their own jurisdiction and authority to enter them, then there is no genuine sphere of independent interpretive authority in the other branches.
Any asserted such sphere of “independent” interpretive authority by others is illusory, and exists solely as a matter of judicial grace. The power to render judgments that bind the actions of other branches swallows up any pretense of co-equal interpretive power. (It also renders any claim that such a judicial power is itself merely a “co-equal” interpretive power a complete misnomer. As Orwell might have put it: all branches possess co-equal interpretive power, but the judiciary’s co-equal interpretive power is more co-equal than everybody else’s.)