I have heard of tribunals awarding residents permanent use of chunks of common property on the grounds that they have been using it for years anyway – I don’t know if that counts as adverse possession or not.
The problem is that, while this hasn’t been a problem so far, it doesn’t mean it won’t be in the future.
My thought would be to have a polite chat with the owner to ask them if they would like to lease the property and then impose a few conditions on the deal, such as right of entry in the event of work needing to be done on common property, or a commitment to its upkeep to an acceptable standard.
I think if this was approached from the point of view that nothing needs to change except that the ownership of the area needs to be stablished, that might get a positive response.