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The $α$ 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 $α$ 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 $α$ Cen AB. We seek to confirm the presence of a convective core in $α$ 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 $α$ 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.