What does that mean? You can have MWI without Quantum Gravity. It just won't have any gravity. If I had to guess, I'd say that you mean that you won't be able to get general relativity working just by doing quantum physics on a non-flat spacetime. You have to have the spacetime metric itself vary along different universes. This is true, and it seems pretty obvious. If you didn't do that, then gravity would have to be the same in all universes. But there's another universe where Earth is somewhere else, so the gravitational field obviously has to be moved.

That link doesn't work for me. Is there somewhere else to get whatever it's intended to link to, or a summary, or something? I find it very difficult to imagine what could possibly constitute an experimental demonstration that Everettian QM requires full quantum gravity and not QM + some semi-classical treatment of gravity. This isn't code for "I don't believe you" -- just a remark that what you're claiming is really startling, at least to me. (Well. In some sense any understanding of QM requires full quantum gravity, in that without it we know we don't have a theory that actually describes the real world. But that's as true of any other theory as it is of Everett's.)