Let me warn you that this open problem is more open ended than the one on Thompson Sampling!
Grothendieck’s inequality and its proof by Gaussian projection give a universal rounding scheme for semidefinite programs. The question is whether there exist a generalization of Grothendieck’s inequality to hyperbolic programs, and what would the rounding scheme look like in this case?
– See section 7 here for a clean description of hyperbolic programming and the associated basic concepts.
– See section 2 here for a simple proof of Grothendieck’s inequality.
– Known connections between hyperbolicity and probability will probably be useful here, see this survey by Robin Pemantle.