I don't know if Windows makes a difference between touchscreen-clicks and mouse-clicks, but if so, it would make sense to get a popup showing up the entries when touching a multi-button.
The popup should be configurable either on long press or immediately on single-click.
I would avoid three-buttons entirely with a touch-based setup. (There aren't any in the default configuration so you'd only get them by choice.) Menus probably make more sense.
I have two/three buttons, because I like to have same layout on all machines (you get used to things after all the years using DO),. And a menu req. one click more on mouse and it's also a space thing. Long press on two buttons also is not best, because of timing (mostly touching longer to avoid single click).
But nevertheless there seems to be no interest in doing this, no prob, was just an idea.