EDIT, 2/25, 7 PM: I'm no longer confident that the claims in this entry are correct. I've found a serious mistake in my proof. It seems like fixable, but I haven't succeeded so far.
EDIT, 2/27, 1:30 AM: THIS ENTRY AS ORIGINALLY POSTED IS FALSE. I've edited to be true, but, uh, much less interesting. False statements are in strikethrough, added
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