6/6 is out of the question. It’s more an optics issue for Disney. It’s better for UOR to go first. People expect to have a fun time at Universal. That’s really it. If they go and some people get sick, it will have been expected and not a huge deal.
Disney is expected to transport guests away from the real world and protect them from all danger. If (when) folks get sick, it will make national headlines and damage the brand. Think of the gator attack. Many viewed it as unacceptable that a wild animal would attack someone in the pixie dust bubble.
So by not rushing to go first, they can more reasonably say they both waited longer than government guidelines, did more than the government required, and cared more about guest health than their competitor.
The calculus for Disney is far more complex than for Universal. Open too early, and they view their guests as dollar signs. Wait too long (especially once guests can visit other parks), and you are spitting in the faces of your biggest fans. This points to all four parks reopening within the next 8 weeks.