Bus to Landmannalaugar late May/ early June

I want to hike from Landmannalaugar to Þórsmörk between 28/05 to 03/06. From what I could gather roads there might not be passable yet. We will have a 4x4 car, but do not want to travel twice so one way of the hike we would want to be driven and then walk down to the car. This website suggests that bus will not begin running until 12th of June. What is the likelihood of the road being open those dates and what alternative bus companies can we find?

Thanks