Check out Cardano's work (their Ouroborous papers). I believe the answer is no. Ironically, they're currently centralized, but not in the sense you're asking about.
Not true, they make the same assumption. They assume that the initial stake of the stake holders is correct. Once everyone agrees on this, then from there on out it's secure. But Proof of Stake doesn't help you come to this agreement initially.
Even Vitalik agrees with me on this one. His so called "weak subjectivity".