There is an intersection in a carpark that I pass through a trillion times a week.

As you can see in the image, its a 4-way with stop lines.
Generally, lane #1 goes straight and #2 goes right, as I've shown. This is the case I'm referring to with this question.
If theres nobody passing vertically, obviously whoever gets to the intersection first at #1 or #2 goes first.
However, when theres vertical traffic, cars end up stopped at #1 and #2 at the same time.
Who has the right-of-way when the vertical traffic clears?
No matter what the law is, everybody seems to think differently. I frequently get into stalemates at this intersection where nobody goes. Or where both go at the same time (and then both stop half-way through, leaving them at the mercy of any new vertical traffic).
The fastest way to make the traffic start flowing seems to be for me to just take the lead and go straight away.
But, this means if the other person does, you both go at eachother. I'm worried about getting into a road rage incident.