What about using a favicon with a different background color for discourse.gohugo.io? It might be one of the other colors in Hugo brand icon, that is blue, green or yellow. I would choose green or blue.
Why?
Because I often find myself with - say - 3 Hugo docs tabs open and 3 Hugo Discussion tabs open (or maybe more) and colors would help me jumping to the right tab.
Just a 2 cent idea… but it takes a few minutes 