  Excess near-infrared emission is detected around one fifth of main-sequence
stars, but its nature is a mystery. These excesses are interpreted as thermal
emission from populations of small, hot dust very close to their stars (`hot
exozodis'), but such grains should rapidly sublimate or be blown out of the
system. To date, no model has fully explained this phenomenon. One mechanism
commonly suggested in the literature is cometary supply, where star-grazing
comets deposit dust close to the star, replenishing losses from grain
sublimation and blowout. However, we show that this mechanism alone is very
unlikely to be responsible for hot exozodis. We model the trajectory and size
evolution of dust grains released by star-grazing comets, to establish the dust
and comet properties required to reproduce hot-exozodi observations. We find
that cometary supply alone can only reproduce observations if dust ejecta has
an extremely steep size distribution upon release, and the dust-deposition rate
is extraordinarily high. These requirements strongly contradict our current
understanding of cometary dust and planetary systems. Cometary supply is
therefore unlikely to be solely responsible for hot exozodis, so may need to be
combined with some dust-trapping mechanism (such as gas or magnetic trapping)
if it is to reproduce observations.
