From early hacking, it looks like registers aren’t append only atm (I thought they were). So, it looks like anyone with write access can merge the tree and delete the history.
That makes having a public writable register more like a variable that anyone can change. Unless I’m missing something (quite possible at this stage).
EDIT:
Specifically, someone with write access can run client.get_register(address).write_merging_branches("replace all entries with this").