Can it be either?
In GR the coordinate time (as well as the spatial coordinates) can in principle be discarded from the formulation of the theory without loss of physical content.
In GR Dirac observables are smeared over all of spacetime, therefore Dirac observables are not local with respect to the unphysical coordinates ##t,x^a##.
You are able to specify a phyical meaning of a local region where a measrement takes place however not specified in terms of some coordinates rather in terms of other measurements, usually in terms of matter degrees of freedom representing material reference systems.
In quantum gravity physical locality can be accomodated while coordinate locality is completely lost.
Something often overlooked is while mathematically in LQG they seem to be able to quantise geometry without matter, when it comes to physical observables matter seems indispensable.
Are you saying it's a local or non local quantum field theory? Because if it's a local quantum field theory of quantum gravity, it has to be wrong.
At present, there is no theorem forbidding gravity from being a local QFT.
There is an offshoot of LQG called group field theory, with more recent developments called tensor field theory. Mathematically they are related to QFT, but in physical interpretation, they are likely to be nonlocal.
Random Tensors and Quantum Gravity
Colored Tensor Models - a Review
Razvan Gurau, James P. Ryan
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The complete 1/N expansion of a SYK--like tensor model
According to Lubos Motl, there is.
Here is a quote from his website concerning asymptotic safety in quantum gravity. He goes into why it can't work because it is a local QFT of quantum gravity...
"There has been a lot of other progress in the recent decade or so. I think that we have realized not only string theory works but even if you assumed that the right theory of quantum gravity is something else, many of the qualitative properties of string theory are necessary for any consistent theory of quantum gravity. We have a much better understanding for the detailed reasons why string theory is the only possible consistent theory of quantum gravity.
Quantum gravity cannot be described as a local field theory in the bulk because of many reasons, including
the infinitely many types of terms that could be added; Weinberg discussed some partial successes but I don't think that there exists any known sensible UV fixed point for gravity; after all, its non-existence was the reason why so many people began to look at Hořava's non-relativistic extension of it recently;
the wrong scaling of the entropy: scale-invariant field theories always have a volume-extensive entropy density and it seems impossible to guarantee that the entropy bounds will be imposed, i.e. that the black hole with its area-extensive entropy remains the record-holder for the total entropy in a volume (and therefore the ultimate stage of a collapse);
the information preservation during the Hawking radiation that implies that physics of quantum gravity must allow for some kind of nonlocal effects that are able to get the information out of the black hole; these effects are impossible if the causal structure dictated by a metric tensor (quantum field) strictly holds;
wrong trans-Planckian, very high-energy scattering amplitudes; the probability to create two particles in such a collision should exponentially decrease, as seen from general black hole thermodynamics, but that won't happen in a local theory in the bulk that is scale-invariant in the UV; the latter would lead to power laws."
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