Nanotechnologies, including molecular quantum dot cellular automata (QCA), are susceptible to high error rates. In this paper, we present the design of concurrently testable latches (D latch, T latch, JK latch, and SR latch), which are based on reversible conservative logic for molecular QCA. Conservative reversible circuits are a specific type of reversible circuits, in which there would be an equal number of 1's in the outputs as there would be on the inputs, in addition to one-to-one mapping. Thus, conservative logic is parity-preserving, i.e., the parity of the input vectors is equal to that of the output vectors. We analyzed the fault patterns in the conservative reversible Fredkin gate due to a single missing/additional cell defect in molecular QCA. We found that if there is a fault in the molecular QCA implementation of Fredkin gate, there is a parity mismatch between the inputs and the outputs, otherwise the inputs parity is the same as outputs parity. Any permanent or transient fault in molecular QCA can be concurrently detected if implemented with the conservative Fredkin gate. The design of QCA layouts and the verification of the latch designs using the QCADesigner and the HDLQ tool are presented.