I would say "watch this space" for these three species. The biggest hurdle with the blackbuck and nilgai has been trying to figure out how to most efficiently manage these herds without major boom and bust cycles. Both blackbuck and nilgai are prolific breeders (blackbuck females can reproduce within their first year of life, and nilgai can regularly twin and triplet), so it is very easy to achieve an overpopulation of both species (which is how the Park ended up with close to 200 blackbuck in the mid 2010s). Nilgai then come with the added difficulty of the males tending to be very aggressive, towards other species and humans. The Park realizes that the Asian exhibits are quite empty nowadays, and there is a push by the curatorial staff to try and save floundering populations, so there is interest in maintaining these species, it is just figuring out how.
To be fair, there has yet to be a gemsbok birth, either, they just have a breeding bull in the herd (hopefully there will be calves this year!). The scimitar-horned oryx herd was also bred around the same time, so there should be scimitar-horned oryx calves soon, as well. The Park has been desperately trying to breed their fringe-eared oryx herd (moving all of the breeding age females behind-the-scenes), but from what I understand, they have been having a very difficult time finding a healthy, fertile breeding bull.
The Park does also intend on breeding the common eland and banteng again; however, they have similarly had a very difficult time sourcing a banteng bull. There are apparently very few breeding-age banteng bulls available in North America right now. There is a young cohort of bulls, but they would be too young to breed the Park's cows.
The bachelor warty pigs were moved to the Park to free up space for the Zoo to breed again. There has ben reinvigorated interest in trying to salvage the population, so the Zoo should be receiving a new male soon, and one of the males from the Park has moved out to be a breeder somewhere else.
The pudu have moved back to their exhibit; however, yes, all of the other species you have mentioned remain behind-the-scenes. The only other species managed entirely behind-the-scenes are the North Indian red muntjac and common warthog.
The remaining AZA animals are purebred, but that is just the Park's herd, four elderly females at St. Louis, two young females at the Dakota Zoo, and a 1.1 pair at the Utica Zoo, so... I believe the plan is to consolidate the remaining breeding age animals at the Park to try and build the numbers back up.
I think the future of the brow-antlered deer herd is still very much up in the air. There are only 2.4 animals left, but they are all of breeding age, so there is potential there, and I think their best hope is with the current curatorial staff... So we will have to see!
