  Over the past decade the relation between the Balmer-line luminosity of HII
galaxies and the velocity width of the emission lines, the L - {\sigma}
relation, has been painstakingly calibrated as a cosmological distance
indicator with seemingly spectacular results: the Hubble constant and the
energy-density of dark energy obtained using the L-{\sigma} indicator agree
remarkably well with the values from canonical indicators. Since most of the
luminosity of these young compact starburst galaxies is emitted by a few narrow
emission-lines, they can be observed with good precision up redshifts z ~ 7
with JWST, making the L - {\sigma} indicator a potentially unique cosmological
probe. However, the precision of the method remains too low to effectively
constrain the relevant cosmological parameters, notably the equation of state
of dark energy. The scatter of the L - {\sigma} relation is significantly
larger than the random observational errors so we do not have a good handle on
the systematics of the method. In a previous paper we posited that since the
ionizing radiation of these young galaxies fades rapidly over time-scales of
only a few million years, age differences could be the main underlying cause of
the scatter. In this paper we explore several different ways to explain the
scatter of the correlation, but without success. We show that the majority of
HII galaxies are powered by multiple starbursts of slightly different ages, and
therefore that the equivalent widths are not reliable chronometers to correct
the luminosities for evolution. Thus, it is not likely that the accuracy of the
L - {\sigma} distance indicator can be improved in the near future. Since we do
not fully understand neither the systematics nor the underlying physics of the
L - {\sigma} relation, using large samples of distant HII galaxies may or may
not improve the accuracy of the method.
