The researchers specifically tested phone face up vs phone face down and noticed no statistical difference, I believe.
Since smart phones became a thing I've always had them set to vibrate, I don't really see why you need your phone to make a sound (and disturb other people) when you get such things as an Instagram like or a game wanting you to play more. Biggest quality of life improvement was disabling vibrate (so it's silent) for instant messages and SMS (which my phone always notifies me twice about?) - they can wait a few minutes to hours and don't need instant replies.