Classical Cepheids in the Nuclear Stellar Disk: membership via dynamical analysis
(abridged) Four classical Cepheids have been associated to the Nuclear Stellar Disk (NSD), suggesting a burst of star formation 20 Myrs ago. In this work, we analyse all available data for these four Cepheids in order to gauge if they are bona fide members of the NSD or interlopers in the region. This allows to confirm or rule out the star formation burst hypothesised in the literature. Our dynamical analysis is based on an updated MWpotential that includes all components of the inner Galaxy. We conduct a thorough study of possible systematics, taking into account both observational errors and model dependencies. Our results suggest that only one Cepheid likely belongs to the NSD, while the orbits of other two extend well beyond it. Accounting for observational uncertainties, only 23-28 percent of the realisations sourced from the error distributions for the latter two are compatible with the NSD. For the fourth there are no proper motion measurements available and the light curve analysis puts into question its classification as a classical Cepheid. Through a comprehensive dynamical analysis of available data on the Cepheids attributed to the NSD, we gauged their membership to this structure. Out of the three Cepheids analysed in unprecedented depth, GCC-a is likely a member of the NSD while both GCC-b and GCC-c do not seem dynamically linked to it. Factoring in the observational uncertainties, a non-negligible percentage of the realisations for GCC-b and GCC-c are still compatible with the NSD, thus membership cannot be completely excluded for either with current data. The robust association of only one Cepheid casts doubt on the occurrence of a star formation episode in the NSD, about 20 Myrs ago.