The Pyra button (and Pandora button for that matter) are supposed to be 'Super' buttons akin to shift, alt, ctrl and Fn.
In the above two examples there are two functions described:
press and release = menu
press and hold = task management
But - the button has the theoretical capability of being used as a modifier for any key on the system.
In the keyboard design, the Pyra button is 'left super' and the lid switch is 'right super' (though I may have left/right backwards on that). The idea there is that doing so allows the lid switch to act as a shift type key - which means that the shoulders can have one use/function with the lid open and another function with the lid closed.
Personally, I had planned to use the Pyra button as an additional programmable key macro layer. Essentially tie Pyra+G to input/fill "Grench" or "sshfs myname@servername:~/myname" and similar type bindings.
Maybe someone could write an application where Pyra+O launches an on-screen overlay and Pyra+K toggles the keyboard to/from the overlay? Either should be captured as keyboard codes.