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Interesting. I wonder if it applies to fresh water fish? I have heard about the leave the hook in for years. Sounds like a better episode for Myth Busters.
If you cut it off as far down as you can, they'll be just fine as long as the hook didn't rip their gill apart. I've had bass swallow my hooks in tournaments and I cut it off, when I got to weigh-in there were all 3 partial hooks laying in my livewell. Don't know how they did it, but they did.