Before revision:  This work focuses on the theoretical design regime of the dynamic mechanism design when the agent makes coupled decisions of reporting and stopping. A dynamic incentive compatibility constraint is introduced to guarantee the robustness of the mechanism to the agent's strategic manipulation. A sufficient condition for dynamic incentive compatibility is obtained by constructing the  payment rules in terms of a set of functions parameterized by the allocation rule.
After revision:  We define the incentive compatibility in this dynamic environment in terms of Bellman equations, which is then simplified by establishing a one-shot deviation principle. Given the optimality of the stopping rule, a sufficient condition for dynamic incentive compatibility is obtained by constructing the payment rules in terms of a set of functions parameterized by the allocation rule.