F-35B nuclear carriage raises weapons-bay and stealth questions

Is the F-35B's lack of a nuclear role caused by the aircraft or by reluctance to carry nuclear weapons on the carriers?

The carrier magazines are recalled as nuclear-certified, while the F-35B is thought to need external carriage that would compromise stealth.

The Queen Elizabeth-class magazines are recalled as being certified for nuclear weapons, but this is explicitly a statement from memory. The F-35B's internal bay is thought unlikely to accommodate the relevant weapon. External carriage is suggested as the alternative, at the cost of stealth. The explanation leaves both the certification recollection and the proposed bay restriction unconfirmed.