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To the first one, Hezekiah's Signet Ring is a Restrict, so a CBN search will still be stopped by it.For the second one, unless something has changed House Hold Idols is not a restrict, but a combination of Ignore and Prevent. So it is a little more complicated. A CBN banding ENHANCEMENT will work if HHI is up, however HEROES with banding abilities cannot enter battle, due to being ignored (don't ask for the specifics of how they are being ignored with no characters ignoring them, it's a little confusing, but that's how it works).
Heroes may not band. Good banding cards have no effect.
household idols kinda sounds like immunity to me. That's just my opinion.