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Website fingerprinting (WF) attacks, usually conducted with the help of a machine learning-based classifier, enable a network eavesdropper to pinpoint which web page a user is accessing through the inspection of traffic patterns. These attacks have been shown to succeed even when users browse the Internet through encrypted tunnels, e.g., through Tor or VPNs. To assess the security of new defenses against WF attacks, recent works have proposed feature-dependent theoretical frameworks that estimate the Bayes error of an adversary's features set or the mutual information leaked by manually-crafted features. Unfortunately, as state-of-the-art WF attacks increasingly rely on deep learning and latent feature spaces, security estimations based on simpler (and less informative) manually-crafted features can no longer be trusted to assess the potential success of a WF adversary in defeating such defenses. In this work, we propose DeepSE-WF, a novel WF security estimation framework that leverages specialized kNN-based estimators to produce Bayes error and mutual information estimates from learned latent feature spaces, thus bridging the gap between current WF attacks and security estimation methods. Our evaluation reveals that DeepSE-WF produces tighter security estimates than previous frameworks, reducing the required computational resources to output security estimations by one order of magnitude.