Convective mixing in Helium-core-burning (HeCB) stars is one of the outstanding issues in stellar modelling. The precise asteroseismic measurements of gravity-modes period spacing ($DeltaPi_1$) has opened the door to detailed studies of the near-core structure of such stars, which had not been possible before. Here we provide stringent tests of various core-mixing scenarios against the largely unbiased population of red-clump stars belonging to the old open clusters monitored by Kepler, and by coupling the updated precise inference on $DeltaPi_1$ in thousands field stars with spectroscopic constraints. We find that models with moderate overshooting successfully reproduce the range observed of $DeltaPi_1$ in clusters. In particular we show that there is no evidence for the need to extend the size of the adiabatically stratified core, at least at the beginning of the HeCB phase. This conclusion is based primarily on ensemble studies of $DeltaPi_1$ as a function of mass and metallicity. While $DeltaPi_1$ shows no appreciable dependence on the mass, we have found a clear dependence of $DeltaPi_1$ on metallicity, which is also supported by predictions from models.
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