Atyy spotted a 1979 paper of Smolin which suggests that a version asymptotic safe QG could exhibit the asymptotic conformal symmetry
required by Nicolai---criterion #4.
Such an approach, only sketched in Smolin's paper of 30 years back, if it could be implemented and found consistent, would meet all four criteria listed here. In that case, here is how the four questions would be answered.
1. Does the approach come to terms with renormalization and the running of couplings?
Yes in fact it is based on that, as an asymsafe QG approach.
2. Does it have spontaneous dimensional reduction, as discussed by Carlip?
One would suppose yes, since this feature has been shown for asymsafe QG. But it would have to be checked for this particular version.
3. Does it have a concrete mathematical realization of 4D spacetime?
Yes it establishes spacetime as an essential frame. Asymsafe QG is a basically a form of General Relativity quantized with running couplings. So it lives on a 4D differential manifold where, however, weird stuff is allowed to happen if you zoom in to very very small scale.
Everything we know and love can still live on the differential manifold as usual. So it is nice and straightforward about that.
4. Could it be adapted so as to fill the bill for Nicolai? Could it acquire conformal symmetry in the limit?
YES! Happily enough the approach has the asymptotic conformal feature that it seems several people are currently interested in. Meissner and Nicolai could run their extreme minimalist version of the Standard Model all the way out to Planck scale on this 4D quantum continuum.
However the M&N standard model extension makes falsifiable predictions. It predicts something which Nicolai says is within reach of the LHC at design energy to rule out.
So one would have a falsifiable package of an SM version build on an AS spacetime, which predicts things about particle mass signatures that can be presently falsified if they aren't true.
That's a hypothetical case, assuming that the AsymSafe QG version that Atyy fished up from 1979 could actually be consistently worked out and all the parts fit. I'm happy. I was wondering about that.
So now we could evaluate all halfdozen 4D QG approaches on our poll, in terms of the same four criteria---the same four questions.
In case anyone wants to look some of this up, here is a thread Atyy started about that 1979 paper:
Here is an online PDF of the paper itself:
Here is that Nicolai talk where he explains why he wants a QG with asymptotic conformal symmetry.
He wants it so as to complete the construction of a minimalist Standard Model able to go to Planck scale without breaking down, which incidentally could be wrong. Nature might tell us she didn't like it but nice try. Such testability is all to the good. Seems like a worthwhile undertaking.