I think you are right. It is probably a handkerchief. He would have hid it better if he was cheating. I wouldn't put cheating past him, but I will be surprised if this goes anywhere.
If it was a handkerchief his campaign would have said so already. I think they know Fox has the video and the second they issue a blanket denial, Fox hits them with the goods.
It very well could have been a hanky, but could it have had some writing on it ... points to make sure he brought up?
He still broke the debate rules.