score:4
It's very likely the roads won't be open yet, according to the Icelandic authority, which publishes statistics on such matters. The median date for opening Landmannalaugar is June 17, the earliest date being May 31 so you're short on luck it seems.
If the roads are closed, I don't think any bus company will attempt to go there. I've checked a few bus companies (here, here, here and here) and all seem to start operating around mid-June, Wikivoyage says the same.
As an aside I recommend this website for checking road and weather conditions in Iceland, it's extremely important you don't underestimate Icelandic weather when driving there.