We investigate the concentration fluctuations in a simple model of electrolyte (two positively and negatively charged Lennard-Jones spheres in a solution of an uncharged Lennard-Jones liquid) confined between electrodes formed by parallel graphene layers. Using constant potential molecular dynamics simulations and extensive constant charge simulations the effect of the proximity to the demixing transition on the electric double layers is analyzed and compared to the results of our continuum mean field theory. In agreement with our previous theoretical findings, we observe a considerable enhancement of the capacitance when temperature is lowered approaching the demixing transition for dilute ionic solutions. This enhancement is less visible when the ionic concentration is increased. Moreover, we observe the new shape of the capacitance, with a maximum at potential of zero charge, and symmetric maxima for positive and negative voltages.