[palmer]

Ties between the executive and judicial branches are determined by the Congress through proper legislation.

Not sure what that means. The executive can act on explicit or implicit powers in the Constitution, or by legislation. The courts have to decide what is Constitutional, or implied, or legislated.

OTOH, there's nothing in the National Emergencies Act about allowing the courts to decide what is an emergency and what is not. That is entirely up to the President and Congress. Congress would need enough votes to override a veto if they want to terminate the emergency. But the emergency has to be declared on some legal authority. In the past the authority was somewhat circular: e.g. https://www.law.cornell.edu/uscode/text/10/2808 used by GWBush after Sep 11. Bush had already declared the emergency by EO: https://georgewbush-whitehouse.archives.gov/news/releases/2001/09/20010914-4.html and several more orders after that to help sieze assets and stop terrorist financing.

[Uncle Sham]

Co-Equal means what it says. The Executive Branch gets to have it's opinion as well. If it's opinion is different from that of the Judicial branch, there is NOTHING the Judicial branch can do to impose it's opinion other than make rulings. If the issue is important enough, Congress can legislate an opinion to solve the issue but they don't necessarily have to address it at all. In this case, the Executive branch, being the enforcement branch can do what it damn well pleases.

Remember, the courts are LIMITED to that which is brought before them. They cannot rule willy nilly just because they have an opinion on something, especially if that something hasn't been brought to them for a ruling.