Conversational recommendation systems (CRS) could acquire dynamic user preferences towards desired items through multi-round interactive dialogue. Previous CRS mainly focuses on the single conversation (subsession) that user quits after a successful recommendation, neglecting the common scenario where user has multiple conversations (multi-subsession) over a short period. Therefore, we propose a novel conversational recommendation scenario named Multi-Subsession Multi-round Conversational Recommendation (MSMCR), where user would still resort to CRS after several subsessions and might preserve vague interests, and system would proactively ask attributes to activate user interests in the current subsession. To fill the gap in this new CRS scenario, we devise a novel framework called Multi-Subsession Conversational Recommender with Activation Attributes (MSCAA). Specifically, we first develop a context-aware recommendation module, comprehensively modeling user interests from historical interactions, previous subsessions, and feedback in the current subsession. Furthermore, an attribute selection policy module is proposed to learn a flexible strategy for asking appropriate attributes to elicit user interests. Finally, we design a conversation policy module to manage the above two modules to decide actions between asking and recommending. Extensive experiments on four datasets verify the effectiveness of our MSCAA framework for the MSMCR setting.