  Low-mass stars in the He-core-burning phase (HeCB) 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 (DP) during the HeCB phase,
provides us with a decisive test to discriminate between competing models of
these stars. We use the MESA, BaSTI, and 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 DP depends strongly on the prescription adopted.
Moreover we show that the detailed behaviour of DP shows the signature of sharp
variations in the Brunt-Vaisala 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.
