Characterize the effect of the Robin boundary function on the theory
Characterize how the qualitative structure of the boundary theory depends on the function $F(\Phi_b)$ specifying generalized Robin boundary conditions, including which features of $F$ determine whether the theory has finite maximal energy and entropy or unbounded energy and entropy.
References
It would be interesting to understand what is the significance of this fact, and in particular to understand how the qualitative structure of the theory depends on the choice of the function $F$. For example, in the special cases we studied, in some the construction led to a theory with a finite maximal energy and entropy, and in others those were infinite. It would be interesting to understand what features of $F$ control this behavior.
One can also consider more exotic situations, where $\rho_b>\rho_+$ holds between two finite values of $\rho_+$, so that there is both a minimal and a maximal energy. More generally, one can consider situation where this equation is satisfied in a union of several line segments. We leave the study of such theories to future work.
It would be interesting to generalize their construction to our setting, where AdS$_3$ is replaced by ${\cal M}_3$, and in particular understand the role of the function $F(\Phi_b)$ in that language.