  The idea of a role for DSR (doubly-special relativity) in quantum gravity
finds some 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
$\kappa$-Poincar\'e Hopf algebras (and of the related $\kappa$-Minkowski
noncommutative spacetime) for the construction of a DSR theory have been
discussed extensively, but a few stubborn 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 key
elements of these 2+1-dimensional models, including the description of
observers, might be incompatible with the DSR principles.
