When it comes to sexual relationships and cohabitation among consenting adults, Utah takes a permissive approach. If a guy wants to shack up with a lady, that's fine. If he wants to shack up with several, no problem. He can father children by different roommates, with no fear of the law.
But if he marries one woman and represents three others as his "spiritual wives," like Kody Brown? Then he's committed a felony. Not because of the stuff that goes on behind closed doors. It's the public act of claiming to be part of a lifelong "plural marriage" that raises the specter of jail...
In challenging the law, they can cite implicit support from the Supreme Court. In a 2003 decision striking down a Texas ban on homosexual sodomy, Justice Anthony Kennedy granted a wide berth to intimate relationships.
I don't really see what Lawrence v. Texas has to do with this. Or rather, citing that case seems completely superfluous. Utah's law literally bans a man from referring to more than one woman as his "spiritual wife". Isn't that speech? And doesn't the very concept of a "spiritual wife" fall squarely under the purview of "free exercise"? That's a First Amendment slam-dunk, two times over.
Presumably polygamists have been reluctant to take this approach because they'd rather just stay off the state and federal governments' radars. And given their history, I can't say that I blame them. But still...it seems like this wouldn't be a terribly difficult case to argue.