  The $\alpha$ Cen stellar system is the closest neighbour to our Sun. Its main
component is a binary composed of two main-sequence stars, one more massive
than the Sun and one less massive. The system's bright magnitude led to a
wealth of astronomical observations over a long period, making it an appealing
testbed for stellar physics. In particular, detection of stellar pulsations in
both $\alpha$ Cen A and B has revealed the potential of asteroseismology for
determining its fundamental stellar parameters. Asteroseismic studies have also
focused on the presence of a convective core in the A component, but as yet
without definitive confirmation. Progress in the determination of solar surface
abundances and stellar opacities have yielded new input for stellar theoretical
models. We investigate their impact on a reference system such as $\alpha$ Cen
AB. We seek to confirm the presence of a convective core in $\alpha$ Cen A by
analysing the role of different stellar physics and the potential of
asteroseismic inverse methods. We present a new series of asteroseismic
calibrations carried out using forward approach modelling and including updated
chemical mixture and opacities in the models. We then complement our analysis
with help of recent asteroseismic diagnostic tools based on inverse methods
developed for solar-like stars. The inclusion of an updated chemical mixture --
that is less metal-rich -- appears to reduce the predicted asteroseismic masses
of each component. Neither classical asteroseismic indicators such as frequency
ratios, nor asteroseismic inversions favour the presence of a convective core
in $\alpha$ Cen A. The quality of the observational seismic dataset is the main
limiting factor to settle the issue. Implementing new observing strategies to
improve the precision on the pulsation frequencies would certainly refine the
outcome of asteroseismology for this binary system.
