A:Answer TL; DR: I tried and the answer is "No."
I have this camera, and I use it to livestream to Twitch. Currently I capture the feed from the camera's HDMI output port to an HDMI capture card. If you try to hit record while capturing the HDMI output the camera puts out the error, "Cannot output during 4K movie shooting." This turns off the capturing and begins the recording. So, no, I don't think its possible.