Ok, now make byobu do that since it is how I actually use tmux. Also make sure it happens on every system I use. This gets tedious and annoying very fast. Heck mouse support not being on by default, requiring a single line in a config file on every system is already annoying enough.
The problem isn't that it is possible in theory. It is annoying in practise, unless you only have one machine and only need to do all this once in one system.