Context. Apparent period variations detected in several eclipsing, close-compact binaries are frequently interpreted as being caused by circumbinary giant planets. This interpretation raises the question of the origin of the potential planets that must have either formed in the primordial circumbinary disk, together with the host binary star, and survived its evolution into a close-compact binary or formed in a post-common-envelope circumbinary disk that remained bound to the post-common-envelope binary (PCEB). Aims. Here we combine current knowledge of planet formation and the statistics of giant planets around primordial and evolved binary stars with the theory of close-compact binary star evolution aiming to derive new constraints on possible formation scenarios. Methods. We compiled a comprehensive list of observed eclipsing PCEBs, estimated the fraction of systems showing apparent period variations, reconstructed the evolutionary history of the PCEBs, and performed binary population models of PCEBs to characterize their main sequence binary progenitors. We reviewed the currently available constraints on the fraction of PCEB progenitors that host circumbinary giant planets. Results. We find that the progenitors of PCEBs are very unlikely to be frequent hosts of giant planets (?10 per cent), while the frequency of PCEBs with observed apparent period variations is very high (~90 per cent). Conclusions. The variations in eclipse timings measured in eclipsing PCEBs are probably not caused by first-generation planets that survived common-envelope evolution. The remaining options for explaining the observed period variations are second-generation planet formation or perhaps variations in the shape of a magnetically active secondary star. We suggest observational tests for both options.
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