I leave the lid off and I have quite a few soaps. The fragrance seems to survive. Indeed, I have a few soaps that are decades old (Paisley, Lenthétic, and Yardley), and the Yardley is in a wooden tub with a loose-fitting lid. Despite being 30 years old or so, the fragrance is quite strong when I lather. So I don't worry about it. Still, no one will know whether you use the lid or not.
The point is simply to let the soap dry, and once it's dry, it's dry. I don't see that leaving it uncapped is a problem.
HOWEVER, my asses' milk shaving soap from Planete Rasoir (my wife brought it back from Paris for me) does, it turns out, lose it's virtues if left unlidded between shaves. I have this from a friend in Quebec, who told me the sad story. It's a terrific soap, and on that one I am careful to replace the lid, based on his experience. But with the Martin de Candre: no lid, no problems.