The first thing that seems a little silly to me about this paper is that he's worrying about interpretations of a theory of quantum gravity, but we don't have a theory of quantum gravity. Even in standard quantum mechanics, the interpretations are not testable, are not needed for any application of quantum mechanics, and belong to philosophy, not science. Since they're irrelevant to standard quantum mechanics, there's no a priori reason to imagine that they're relevant to a theory of quantum gravity, which we don't have yet.
Quantum gravity has problems in general with talking about observers. E.g., if your system is the entire universe, you don't have any external observer who can make measurements on it. It's possible that this is of deep significance and holds a clue as to how to construct a theory of quantum gravity. It's also possible that it's completely unimportant, just as observers are an inessential concept in standard QM.
The standard presentation of MWI assumes perfect linearity, which explains why the parallel universes can't influence one another. This lack of "cross-talk" is what makes it impossible to test MWI against CI empirically. GR is nonlinear, so it's reasonable to assume that a hypothetical theory of quantum gravity would be nonlinear. The natural inference seems to me to be not that this is a problem for MWI but that it conceivably makes MWI testable.
Singh talks about the hole argument, and that whole part of the paper seems extremely flaky to me. The classical hole argument is resolved by understanding what is and isn't an observable in GR. Now he's trying to apply it to quantum gravity, but he doesn't make any effort to consider the issue of what is and isn't an observable in quantum gravity.
He talks about specific toy models of nonlinear quantum mechanics, apparently with the motivation of associating their nonlinearity with the CI's wavefunction collapse. But it's far from clear to me that this association holds. In a nonlinear quantum mechanics, you're going to get all kinds of new phenomena. There is no reason to think that out of all these phenomena, somehow something will pop out that acts like CI with wavefunction collapse as an actual physical process.