According to this news story, the state chair of the South Carolina Republican Party said the party will soon file a lawsuit to win a closed primary for itself.
The party might lose such a lawsuit, however. The Virginia Republican Party already had a similar lawsuit, and the 4th circuit said the party only has a right to insist on a closed primary for itself in instances at which the party has no choice but to nominate by primary. Both Virginia and South Carolina are in the 4th circuit, so the Virginia precedent will control. The Virginia case is Miller v Brown, 462 F 3d 312 (2006). South Carolina law says any qualified party is free to nominate by convention or primary. The party decides. Major parties in South Carolina always choose a primary, instead of a convention, but because the party is legally free to choose a convention, it will have difficulty winning the lawsuit.
One of the ironies is that the South Carolina state Republican chair made this promise to a Tea Party meeting. Yet in Texas, which also has open primaries, polls have shown that the 2010 gubernatorial candidate favored by Tea Party activists, Debra Medina, receives more popular support from self-described independent voters than she does from self-described Republican voters.