Suppose that a convex-operational theory has two properties: tomographic locality (states on composite systems are uniquely characterized by the statistics of local measurements) and transitivity (every two pure states are connected by a reversible transformation). Does it follow that this theory is automatically a subtheory of quantum mechanics? This conjecture has first been formulated by Tony Short. In the talk, I give evidence for this conjecture, but also for a surprising way in which it could fail: namely, by going from pairs to triples of systems.