Is it possible to have the Steam Link application automatically connect to the paired computer on boot?

I am already using a systemd service to start the Steam Link application itself, and it works fine, but it requires a click/button press to connect to my computer, which means I need an input device connected to the Pi.

To be clear, I am referring to this screen https://imgur.com/3germ3l which appears on Steam Link startup, which requires me to click "Start Playing" to connect to the computer.

This is not optimal for my setup, I'd rather leave my bluetooth gamepad paired to my computer all the time.