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The chase is a famous algorithmic procedure in database theory with numerous applications in ontology-mediated query answering. We consider static analysis of the chase termination problem, which asks, given set of TGDs, whether the chase terminates on all input databases. The problem was recently shown to be undecidable by Gogacz et al. for sets of rules containing only ternary predicates. In this work, we show that undecidability occurs already for sets of single-head TGD over binary vocabularies. This question is relevant since many real-world ontologies, e.g., those from the Horn fragment of the popular OWL, are of this shape.