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The confluence property of ground (i.e., variable-free) term rewrite systems (GTRS) is well-known to be decidable. This was proved independently by M. Dauchet et al. (1987; 1990) and by M. Oyamaguchi (1987) using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, it has been a well-known longstanding open question whether this bound is optimal. The authors give a polynomial-time algorithm for deciding the confluence of GTRS, and hence alsofor the particular case of suffix- and prefix string rewrite systems or Thue systems. We show that this bound is optimal for all these problems by proving PTIME-hardness for the string case. This result may have some impact on other areas of formal language theory, and in particular on the theory of tree automata.