What experiment would disprove loop quantum gravity?

So, "phenomenological quantum gravity" is the particular field where we consider experimental tests of quantum gravity (with the hopes of winnowing the 3 dozen competing theories down to a handfull). To the best of my knowledge, Giovanni Amelino-Camelia pioneered the field (c.f., Amelino-Camelia's review article).

At any rate, when focusing on loop quantum gravity, there have been a few preprints. Most deal with the phenomology of loop quantum cosmology, but this one appears more general:

• Florian Girelli, Franz Hinterleitner, Seth A. Major, "Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics". Eprint

I think it's precisely what the OP is looking for...

Please correct me if I'm wrong, but I thought that no one had been able to show how General Relativity (GR) emerged from LQG in the semi classical limit ? In this context, it may make sense to ask for experimental manifestations but to me it would seem more important to make sure first that LQG gives in some limit a classical theory of gravitation before all.

String theory, which describes a theory of quantum gravity, treats short distances by means of so called dualities; if you try to probe distances smaller than some scale, you obtain the exact same physics as the one at larger scales (this is called T-duality). So it has a minimal distance, as you would expect somehow from quantum gravity, though it includes it in a continuous manner.

First of all, Loop Quantum Gravity doesn't postulate the granularity of space: it's a logical consequence of the canonical quantization of Einstein's Lagrangian and it's associated whit the spectra of the Volume operator (for mathematical details take a look at these lectures https://arxiv.org/pdf/gr-qc/0210094.pdf). Now, a minimum length doesn't imply that Loop Quantum Gravity violates Lorentz symmetry: it's the same as saying that discrete eigenvalues for the angular momentum violate rotational invariance(!); so you can' t look for Lorentz violations in order to disprove the theory. Anyway, a lot of work has been done in order to derive results which can verify or falsify LQG. In LQC (the application of LQG to Cosmology) the factor scale of the FRW metric doesn't vanish leading to a bounce-phase which can in principle be detected today (see https://arxiv.org/pdf/1308.4348v2.pdf); not only this, applying LQG to black holes shows that they decay whit a life-time, emitting radiations which in principle can be detected. (be careful, it's a different process from Hawking radiation, for more see https://arxiv.org/pdf/1605.05268.pdf)