Kody Brown and wives Meri, Janelle, Christine and Robyn filed a lawsuit in Salt Lake City's U.S. District Court in July. Oral arguments in the case are set for Friday before U.S. District Judge Clark Waddoups.
The stars of the TLC show "Sister Wives" said the law is unconstitutional because it prohibits them from living together and criminalizes their private sexual relationships.
Under Utah law, people are guilty of bigamy if they have multiple marriage licenses, or if they cohabitate with another consenting adult in a marriage-like relationship.
Formerly of Lehi, the Browns and their 17 children moved to Nevada in January after police launched a bigamy investigation. The Browns practice as part of their religious beliefs.
Oral arguments in the case are set for Friday in Salt Lake City's U.S. District Court.
It's not clear when Waddoups will rule.
State prosecutors contend the Browns - who haven't been charged - aren't facing any real harm because the state has rarely prosecuted individuals for bigamy without also prosecuting for other crimes, such as underage marriages, sexual abuse or welfare fraud.
"They have not been warned that if they do not cease to engage in their polygamous relationships that legal actions will be taken against them," Assistant Utah Attorney General Jerrold S. Jensen has said in court papers asking the judge to dismiss the case.
"And - what is probably the tipping point - there have been no arrests or prosecutions for the mere practice of polygamy in Utah in over 50 years."
A check of state court records by The Associated Press found at least two cases, however.
Bob Foster had three wives when he was arrested and charged with bigamy in 1974. He pleaded no contest and was sentenced to six months in Jail. He was released after 21 days and ordered to serve five years of probation. A judge also said Foster was not allowed to live with his families. Foster died from cancer in 2008. He was still married to all three women.
Mark Easterday was arrested and charged with bigamy in 1999. Authorities were alerted to Easterday's multiple marriage by his first wife as part of a custody battle during their divorce. He ultimately pleaded no contest to adultery because the divorce was finalized before the bigamy case went to trial. Easterday was sentenced to probation.
Easterday, who left Utah and is currently married to two women, told The Associated Press he believes the Browns are right to fear a bigamy prosecution.
"I know from experience that they do prosecute," Easterday said. I think they should change the law over the entire country. Why it is that in some places a woman and a woman can be married, but a man can't have another wife?"
Polygamy in Utah and across much of the Intermountain West is a legacy of 19th century Church of Jesus Christ of Latter-day Saints. Mormons abandoned the practice of plural marriage in the 1890s as a condition of Utah's statehood. The church now excommunicates members found engaging in polygamy.
An estimated 38,000 self-described Mormon fundamentalists continue the practice, believing it brings exaltation in heaven. Most keep their way of life a secret. In most cases, polygamous men are legally married to their first wives and marry subsequent brides only religious ceremonies.