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Rapid response, namely low latency, is fundamental in search applications; it is particularly so in interactive search sessions, such as those encountered in conversational settings. An observation with a potential to reduce latency asserts that conversational queries exhibit a temporal locality in the lists of documents retrieved. Motivated by this observation, we propose and evaluate a client-side document embedding cache, improving the responsiveness of conversational search systems. By leveraging state-of-the-art dense retrieval models to abstract document and query semantics, we cache the embeddings of documents retrieved for a topic introduced in the conversation, as they are likely relevant to successive queries. Our document embedding cache implements an efficient metric index, answering nearest-neighbor similarity queries by estimating the approximate result sets returned. We demonstrate the efficiency achieved using our cache via reproducible experiments based on TREC CAsT datasets, achieving a hit rate of up to 75% without degrading answer quality. Our achieved high cache hit rates significantly improve the responsiveness of conversational systems while likewise reducing the number of queries managed on the search back-end.