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It is known for some time that the cosmological bound on the invisible axion scale can be avoided by an early phase of strong QCD. While most approaches rely on theories where the strong coupling constant is determined through the expectation value of some scalar field, we show that this mechanism can also be implemented into the benchmark KSVZ and DFSZ models when the early phase of strong QCD emerges by the modification of the running coupling during inflation. For both models the physics that are responsible for making QCD strong do not displace the axions minimum by too much, so that the efficiency of the relaxation is controlled by parameters of the theory and the number of inflationary e-folds. In particular, we consider the case of very efficient relaxation where the axion abundance is dominated by inflationary quantum and post-inflationary thermal fluctuations. Within this situation we identify the parameter space compatible with all cosmological constrains and derive conditions on the reheating temperature and the QCD scale during inflation that result in the axion making up all the dark matter. Due to duality below the Peccei-Quinn scale and a minor influence of the KSVZ and DFSZ fields on the running, our findings also apply to a minimal 2-form implementation of the axion.