However, after some testing, I have found out that you cannot leash the horse that you have just dismounted if you do not have the vandalism permissions on the claim. However, you can mount your own horse without any permissions. In my point of view, the horse owners should be able to leash their animals without any permissions required.
I cannot assign the vandalism rights, since then it would allow other visitors to steal people's horses or break the hitching posts. Resolving this issue would allow cultural centres and trade hubs to have functional stables.
