This same team (Gisin et al) works around all the various FTL models, including BM. There are so many variants, all I can say is that is that you won't really see most theoretical treatments accompanied by a related experiment. As you already know, variants with FTL<10,000c are already ruled out experimentally. Generally, variants with an absolute reference frame have been ruled out in various manners. But again, most of these type arguments are rejected by supporters anyway.
So... specifically ruling out [10,000c<FTL<infinity] models is not a particularly hot topic. Quantum nonlocality - well-accepted in science today - does not really have a "speed" - infinite or otherwise. Delayed choice experiments make a mockery of there being a speed that can be calculated. Such experiments fit with garden QM with no additional assumptions required. However, there is no explanatory mechanism behind the usual QM.