Based on past experience, I highly doubt they will.
Can't exactly blame the press if the RNC PR department is run by idiots.
The RNC has not exactly done a great job over the past few years, but they can’t touch this stuff and they know it. The news media will never, ever investigate Obama, period. Forget it. All they will do is accuse the Republicans of racism for even bringing it up, and do so loudly. You surely know that any accusation of racism is treated as proof of guilt, and the RNC does not want to have to fight that battle at this point in the campaign. They are just going to have to win on the merits.