If there are no Hives available on the war room map, exit to the main menu and go back in. I did intersperse Forest Hive and Desert Hive (not the Elite versions) whenever they popped up. I did the 'Let the Games Begin' Hive Mission as the main Hive of choice. I did 30 on Challenge 3 and then 20 on Challenge 2 and it still popped up so I can confirm that it works on those difficulties. I was working towards the 'Do 30 war zones on Challenge 3 trophy' at the same time as this. I saw a person say on a YouTube comment that he had done 100 runs thinking the counter was glitched before he realised. Theres been multiple reports of it not working with Challenge 1 runs. You MUST also be on Challenge 2 or higher. Unfortunately you can't get it at the same time, as only the host can make progress apparently. If you are helping a friend work towards the trophy, make sure the person you want it to work for is the host.
So make sure you disable the matchmaking and play solo. You MUST be the host for it to count on the servers. There are a few things that, if you are not in the know about, you are gonna really struggle to get it to work. This is what I did in order to get it to work. I thought I would make this post as I saw quite a few people on Reddit, YouTube and other discussion groups who seemed to be having a hard time getting this one to work.