Context: Our understanding of the s-process nucleosynthesis in asymptotic giant branch (AGB) stars is incomplete. AGB models predict, for example, large overabundances of lead (Pb) compared to other s-process elements in metal-poor low-mass AGB stars. This is indeed observed in some extrinsically enhanced metal-poor stars, but not in all. An extensive study of intrinsically s-process enriched objects is essential for improving our knowledge of the AGB third dredge-up and associated s-process nucleosynthesis. Aims: We compare the spectral abundance analysis of the SMC post-AGB star J004441.04-732136.4 with state-of-the-art AGB model predictions with a main focus on Pb. The low S/N in the Pb line region made the result of our previous study inconclusive. We acquired additional data covering the region of the strongest Pb line. Methods: By carefully complementing re-reduced previous data, with newly acquired UVES optical spectra, we improve the S/N of the spectrum around the strongest Pb line. Therefore, an upper limit for the Pb abundance is estimated from a merged weighted mean spectrum using synthetic spectral modeling. We then compare the abundance results from the combined spectra to predictions of tailored AGB evolutionary models from two independent evolution codes. In addition, we determine upper limits for Pb abundances for three previously studied LMC post-AGB objects. Results: Although theoretical predictions for J004441.04-732136.4 match the s-process distribution up to tungsten (W), the predicted very high Pb abundance is clearly not detected. The three additional LMC post-AGB stars show a similar lack of a very high Pb abundance. Conclusion: From our study, we conclude that none of these low-mass, low-metallicity post-AGB stars of the LMC and SMC are strong Pb producers. This conflicts with current theoretical predictions.