It does not, because this button, like every OTHER button in the mouse, is configurable.
So if you don’t want to change DPI at all for your gaming/work flow – You can just give yourself extra buttons.
On Windows to get any functionality out of a Logitech Mouse’s extra buttons you gotta get Logi’s proprietary programme. On Linux, Piper/Ratbag does a great job, but I just couldn’t find this specific mapping (and additionally when I first plugged in the mouse, meaning it’d be on factory settings, the DPI shift button wasn’t doing anything) and that’s why I’m wondering if it’s unsupported.