Cascaded binary hypothesis testing is studied in this paper with two decision centers at the relay and the receiver. All terminals have their own observations, where we assume that the observations at the transmitter, the relay, and the receiver form a Markov chain in this order. The communication occurs over two hops, from the transmitter to the relay, and from the relay to the receiver. Expected rate constraints are imposed on both communication links. In this work, we characterize the optimal type-II error exponents at the two decision centers under constraints on the allowed type-I error probabilities. Our recent work characterized the optimal type-II error exponents in the special case when the two decision centers have same type-I error constraints and provided an achievability scheme for the general setup. To obtain the exact characterization for the general case, in this paper we provide a new converse proof as well as a new matching achievability scheme. Our results indicate that under unequal type-I error constraints at the relay and the receiver, a tradeoff arises between the maximum type-II error probabilities at these two terminals. Previous results showed that such a tradeoff does not exist under equal type-I error constraints or under general type-I error constraints when a maximum rate constraint is imposed on the communication links.