They breed easily in captivity and reintroduction into reserves is also very successful.
So the answer to the riddle, I think, has to do with the pack size:
The behaviour of Wildogs are characterised by their reliance on 'helpers'. This includes cooperative hunting, defence from competitors for their prey (hyenas), pup feeding and baby-sitting. Because of this there is a strong positive correlation between pack size and the production and survival of pups. Consequently, a pack in which membership drops below a critical size may be caught in a positive feedback loop: poor reproduction and low survival further reduce pack size, culminating in failure of the whole pack (source: "Crucial importance of pack size in the African wild dog" - Courchamp & Macdonald, 2000).
I have seen a small group of four, after being reintroduced from breeding in the Madikwe reserve (that was 2001). This is a guided only reserve (no self-drive). The ranger at the time said that particular pack used to be 5 (adults only). Based on the above paper, I would think their chances of successfully increasing in numbers would have been small. So this poses the challenge of only reintroducing them to reserves once a sufficiently large pack size is reached in breeding. I do not know if this is actually being practised though.