The idea of a role for DSR (doubly-special relativity) in quantum gravity finds encouragement in a few scenarios, but in order to explore some key conceptual issues it is necessary to find a well-understood toy-quantum-gravity model that is fully compatible with the DSR principles. Perhaps the most significant source of encouragement comes from the recent proposal of a path for the emergence of DSR in Loop Quantum Gravity, which however relies on a few assumptions on the results of some computations that we are still unable to perform. Indications in favor of the possibility of using some elements of κ-Poincaré Hopf algebras (and of the related κ-Minkowski spacetime) for the construction of a DSR theory have been discussed extensively, but a few open issues must still be resolved, especially in the two-particle sector. It has been recently observed that certain structures encountered in a formulation of 2+1-dimensional classical-gravity models would fit naturally in a DSR framework, but some elements of these models, including the description of observers, might be incompatible with the DSR principles.
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