We investigate the luminosity suppression and its effect on the mass-radius relation as well as cooling evolution of highly magnetised white dwarfs. Based on the effect of magnetic field relative to gravitational energy, we suitably modify our treatment of the radiative opacity, magnetostatic equilibrium and degenerate core equation of state to obtain the structural properties of these stars. Although the Chandrasekhar mass limit is retained in the absence of magnetic field and irrespective of the luminosity, strong central fields of about $10^{14}, {rm G}$ can yield super-Chandrasekhar white dwarfs with masses up to $1.9, M_{odot}$. Smaller white dwarfs tend to remain super-Chandrasekhar for sufficiently strong central fields even when their luminosity is significantly suppressed to $10^{-16} L_{odot}$. Owing to the cooling evolution and simultaneous field decay over $10 {rm Gyr}$, the limiting masses of small magnetised white dwarfs can fall to $1.5 M_{odot}$ over time. However the majority of these systems still remain practically hidden throughout their cooling evolution because of their high fields and correspondingly low luminosities. Utilising the stellar evolution code $textit{STARS}$, we obtain close agreement with the analytical mass limit estimates and this suggests that our analytical formalism is physically motivated. Our results argue that super-Chandrasekhar white dwarfs born due to strong field effects may not remain so for long. This explains their apparent scarcity in addition to making them hard to detect because of their suppressed luminosities.
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