The prevalence of ubiquitous location-aware devices and mobile Internet enables us to collect massive individual-level trajectory dataset from users. Such trajectory big data bring new opportunities to human mobility research but also raise public concerns with regard to location privacy. In this work, we present the Conditional Adversarial Trajectory Synthesis (CATS), a deep-learning-based GeoAI methodological framework for privacy-preserving trajectory data generation and publication. CATS applies K-anonymity to the underlying spatiotemporal distributions of human movements, which provides a distributional-level strong privacy guarantee. By leveraging conditional adversarial training on K-anonymized human mobility matrices, trajectory global context learning using the attention-based mechanism, and recurrent bipartite graph matching of adjacent trajectory points, CATS is able to reconstruct trajectory topology from conditionally sampled locations and generate high-quality individual-level synthetic trajectory data, which can serve as supplements or alternatives to raw data for privacy-preserving trajectory data publication. The experiment results on over 90k GPS trajectories show that our method has a better performance in privacy preservation, spatiotemporal characteristic preservation, and downstream utility compared with baseline methods, which brings new insights into privacy-preserving human mobility research using generative AI techniques and explores data ethics issues in GIScience.