To my knowledge, there are still 3 open conjectures concerning simple theories : elimination of hyperimaginaries, Lascar strong type = strong type, and stable forking conjecture (if a type forks over some parameters, this is due to an instance of a stable formula). I haven't think of it very hard, but intuitively elimination of hyperimaginaries implies Lstrong type = strong type, isn't it ? Are there more implications between those 3 conjectures ? On the other hand, do you think the resolution of one of them would be an important breakthrough in model theory ? Or at the contrary would be rather anecdotal ?