That's not the reason they can't. They can't register the trademark because it's a descriptive one.
If I try to trademark "hacker forum", an EU trademark officer will reject it not because my website doesn't have hackers on it, but because it's descriptive and prevents others from starting hacker forums.
So
> They could call themselves... ClosedAI
is also incorrect, because it's descriptive as well.
Not the issue. Per the ruling even if their AI was open they still couldn't have the trademark.