Question is in the title. Jech states in his book on page 696 that the consistency strength is "in the region of Woodin cardinals," which is frustratingly imprecise. I tried to find a reference for such, but couldn't find anything.

I gave my attempt at an inner-model-theoretic approach to a lower bound, but got stuck in the middle of it, and needed some help. As a novice in inner model theory, it would be helpful to see this written out if it's not too crazy. I'd have no idea if it's simply above my pay grade.

For an upper bound, you can certainly overshoot and use Foreman's result that the consistency of a huge cardinal implies that ever regular cardinal can carry a saturated ideal. But I'm wanting something more precise. Foreman's argument doesn't look modifiable to bring this down to the Woodin realm.

Where would I find a reference which discusses this?