Low-mass stars in the He-core-burning (HeCB) phase play a major role in stellar, galactic, and extragalactic astrophysics. The ability to predict accurately the properties of these stars, however, depends on our understanding of convection, which remains one of the key open questions in stellar modelling. We argue that the combination of the luminosity of the AGB bump (AGBb) and the period spacing of gravity modes ({$Δ$}{$Π$}$_{1}$) during the HeCB phase provides us with a decisive test to discriminate between competing models of these stars. We use the Modules for Experiments in Stellar Astrophysics (MESA), a Bag of Stellar Tracks and Isochrones (BaSTI), and PAdova {\amp} TRieste Stellar Evolution Code (PARSEC) stellar evolution codes to model a typical giant star observed by Kepler. We explore how various near-core-mixing scenarios affect the predictions of the above-mentioned constraints, and we find that {$Δ$}{$Π$}$_{1}$ depends strongly on the prescription adopted. Moreover we show that the detailed behaviour of {$Δ$}{$Π$}$_{1}$ shows the signature of sharp variations in the Brunt-V{̈a}is{̈a}l{̈a} frequency, which could potentially give additional information about near-core features. We find evidence for the AGBb among Kepler targets, and a first comparison with observations shows that, even if standard models are able to reproduce the luminosity distribution, no standard model can account for satisfactorily the period spacing of HeCB stars. Our analysis allows us to outline a candidate model to describe simultaneously the two observed distributions: a model with a moderate overshooting region characterized by an adiabatic thermal stratification. This prescription will be tested in the future on cluster stars, to limit possible observational biases.