With or without AJAX, any time you send something the the chat, the box automatically resets to the checked position (Teammates only). I'd rather have the checkbox remain unchecked upon a reload if that is what it was sent before the reload of the page/chat. Sometimes I send multiple lines to the general chat (since there is a character limit per each line sent) and I always forget about the checkbox and so most of it is sent to my teammate(s) only.
Priority: I donno, a 2 at best. It's nothing vital or important, but I think it would make things a bit easier, and, color me ignorant, but I would think it's easy to code.