The rules are unclear as to whether the Sol and Hacan special victory occur during round 8, or after round 8. If Xxcha predicts Hacan win in round 8, and Hacan achieves their special victory, does Xxcha steal the win?

I am surprised not to see this addressed in the FAQ!

It's not addressed in the FAQ because it is addressed in the rulebook. Page 14:

"If the chosen player wins (alone or as an ally, even as the Xxcha’s ally) with any victory condition during the predicted game round, the Xxcha player wins the game instead."

Emphasis FFG's. There is only one victory condition that any player is allowed to qualify for (control of strongholds.) The only additional victory conditions are the Hacan and Sol special victory conditions. So, the fact that Xxcha can predict "any" victory condition would mean that he can in fact steal the victory from Hacan after turn 8.

I hear you. The point of confusion is that the rules make it seem as though the special victories happen after the 8th round.

The rules say that if no one has won by "the end of the 8th round," the special victories may take effect, implying that the check for noramal win marks the end of the round. Plus, from what I understand, the homolog of the Xxcha ability from Dune does NOT take effect on the other special victory conditions.

Probably reading too much into it, but a clarification in the FAQ would be a simple matter.

That is correct, the Bene Gesserit prediction ability in Dune did not allow prediction of special victories. Rex is different in that regard.

If you want an official answer to the question, you can always ask FFG directly.