Product descriptions on e-commerce websites often suffer from the missing of important aspects. Clarification question generation (CQGen) can be a promising approach to help alleviate the problem. Unlike traditional QGen assuming the existence of answers in the context and generating questions accordingly, CQGen mimics user behaviors of asking for unstated information. The generated CQs may help vendors check and fill in those missing information before posting, and improve consumer experience consequently. Due to the variety of possible user backgrounds and use cases, the information need can be quite diverse and specific, while previous works assume generating one CQ per context and the generation results tend to be generic. We thus propose the task of Diverse CQGen and also tackle the challenge of specificity.We propose a newmodel named KPCNet, which generates CQs with Keyword Prediction and Conditioning, to deal with the tasks. Automatic and human evaluation on 2 datasets (Home & Kitchen, Office) showed that KPCNet can generate more specific questions and promote better group-level diversity than several competitive baselines.