Is their Working Path under Edit Horse, set to Explore only? I did look over your horses and the tab you have set for Explore only, all of their energy has been used up. Majority of your horses actually have all of their energy used up except for a few select 2 year olds under Uncategorized Horses tab and some foals. If you mean the ones under Uncategorized, I'd least look to see if they are set to Explore Only for their Working Path and see if that helps first and foremost.

I cannot say for anyone else but any horse I wish to use at Hoofbeat Plains, I explicitly have my horses set to Explore only in their Working Path and that's it.