NO, not according to the Constitution.
The exception (which itself is not constitutional) swallows the (constitutional) rule.
Whatever some court decision(s) have said, the absurdity of such rulings would certainly have been apparent to our Founders, had they ever imagined them in their worst nightmares.
But alas, some things (and court rulings) are so outlandish that nobody can possibly forsee them happening anywhere (other than a communist or nazi dictatorship, or perhaps in a lunatic asylum — and our Founders intended none of these for our national republic).
So far as I can recall on quick memory, the Supreme Court has watered down, undermined, or effectively destroyed much of the 1st, 2nd, 4th, 7th, 8th, 9th, 10th amendments in our Bill of Rights. Judgement calls in some cases, in others pure authoritarian diktat. The result is one constitutional law professor we know quit teaching the subject because she grew weary of always having to explain to students how the court has destoryed or cancelled or undermined or just plain weakened so many of our liberties, even though they were fought for in blood and still exist on paper (anyway).
We are as close to a dictatorship as is possible, and it will only take a few EO’s or Acts of Congress to fully (and probably permanently) nail that coffin shut on our country.
You'd be surprised how much of "your" information you voluntarily give away to others on a regular basis.