In a $D=2+1$ state with edge modes described by a chiral $( c_L \neq c_R )$ CFT there is a predicted thermal Hall conductance associated with the gravitational anomaly at the edge. This is the case for integer and some fractional quantum hall systems. For just free bosons on the edge the conductance is proportional to the difference in number between left and right movers as well as the temperature: $K_H \propto (c_L - c_R) T$
This seems like the sort of thing someone with a lab and a fancy set of probes might be able to measure. What's the state of experimental progress in this direction?
I'm familiar with a paper by Kane and Fisher which describes the difficulty a bit and proposes an experimental set-up but since the paper is almost 20 years old I was curious about the modern state of things. A review of some of the physics that goes into making this measurement would be appreciated!