Cluster expansion and the boxdot conjecture

The boxdot conjecture asserts that every normal modal logic that faithfully interprets T by the well-known boxdot translation is in fact included in T. We confirm that the conjecture is true. More generally, we present a simple semantic condition on modal logics $L_0$ which ensures that the largest logic where $L_0$ embeds faithfully by the boxdot translation is $L_0$ itself. In particular, this natural generalization of the boxdot conjecture holds for S4, S5, and KTB in place of T.