  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.
