Summary: Swendsen-Wang dynamics for the Potts model was proposed in the late 1980’s as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. V. K. Gore and M. R. Jerrum [J. Stat. Phys. 97, No. 1–2, 67–86 (1999; Zbl 1006.82015)] found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with \(q\geq 3\) colors on the complete graph on \(n\) vertices at the critical point \(\beta_c(q)\), Swendsen-Wang dynamics has \(t_{\text{mix}}\geq \exp (c\sqrt{n})\). A. Galanis et al. [LIPIcs – Leibniz Int. Proc. Inform. 40, 815–828 (2015; Zbl 1375.82019)] showed that \(t_{\text{mix}}\geq \exp (cn^{1/3})\) throughout the critical window \((\beta_s,\beta_S)\) around \(\beta_c \), and A. Blanca and A. Sinclair [LIPIcs – Leibniz Int. Proc. Inform. 40, 528–543 (2015; Zbl 1375.60133)] established that \(t_{\text{mix}}\geq \exp (c\sqrt{n})\) in the critical window for the corresponding mean-field FK model, which implied the same bound for Swendsen-Wang via known comparison estimates. In both cases, an upper bound of \(t_{\text{mix}}\leq \exp (c'n)\) was known. Here we show that the mixing time is truly exponential in \(n\): namely, \(t_{\text{mix}}\geq \exp (cn)\) for Swendsen-Wang dynamics when \(q\geq 3\) and \(\beta \in (\beta_s,\beta_S)\), and the same bound holds for the related MCMC samplers for the mean-field FK model when \(q>2\).