If you try to turn down the ambient volume while on a board, it doesn't actually do anything. Even if you turn off the main volume slider, you'll still hear the ambient sounds. There's no way to actually turn off the ambient sound all-together.
I also found this behaviour unexpected. Didn't know I had to change screens for it to take effect and so ended up muting the sound for the entire browser instead (which is suboptimal). Do the developers consider this a bug? Would be nice if the changes took effect immediately or if the limitation was noted in the UI.