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This problem arises often when concerned with long-term treatment effects since experiments are often short-term due to operational necessity while observational data can be more easily collected over longer time frames but may be subject to confounding. In this paper, we tackle the challenge of persistent confounding: unobserved confounders that can simultaneously affect the treatment, short-term outcomes, and long-term outcome. In particular, persistent confounding invalidates identification strategies in previous approaches to this problem. To address this challenge, we exploit the sequential structure of multiple short-term outcomes and develop several novel identification strategies for the average long-term treatment effect. Based on these, we develop estimation and inference methods with asymptotic guarantees. 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