As a gap shooter, shooting with both eyes open screws with my left/right. Since I do consciously see the arrow, having both eyes open gives me two images of the arrow. I haven't trained myself to ignore the left eye image. And there really is no good reason to go through that. Depth perception? So what? I know how I'm going to take the shot before I draw the bow.
You see the reality of it is, which method gives YOU the greatest accuracy? Make a solid objective decision for yourself. Go out and on one day shoot a target with one eye closed. Keep that target. Go out the next day and shoot a new target with the same face pattern but with both eyes open. Same number of shots. (Make your own targets if necessary, paper plate with a spot, whatever.) Compare the results. There ya go.