[grraaaaahhh]

E cannot be correct.

"All my hats are green" is still false even when I own a red hat and a green hat.

[Supermancho]

I would agree that's obvious, if not for the original error.

The liar doesn't necessarily "have" any hats. Again, the assumption that the liar has hats is incorrect because it's relying on an conversational implication, rather than a specific assertion.

[grraaaaahhh]

Sure, but the question isn't about which statement is possible from the liar's statement, it's about which statement we can conclude from the liar's statement.

The liar could be lying because they have no hats. They could be lying because they have a non-green hat. We cannot conclude E because it's possible that E is not correct.