  Context: The nature of the progenitors of type Ia supernova (SN) progenitors
remains unclear. While it is usually agreed that single-degenerate progenitor
systems would be luminous supersoft X-ray sources (SSSs), it was recently
suggested that double-degenerate progenitors might also go through a SSS phase.
Aims: We aim to examine the possibility of double-degenerate progenitor systems
being SSSs, and place stringent upper limits on the maximally possible
durations of any SSS phases and expected number of these systems in a galactic
population. Method: We employ the binary population synthesis code SeBa to
examine the mass-transfer characteristics of a possible SSS phase of
double-degenerate type Ia SN progenitor systems for 1) the standard SeBa
assumptions, and 2) an optimistic best-case scenario. The latter case
establishes firm upper limits on the possible population of supersoft source
double-degenerate type Ia SN progenitor systems. Results: Our results indicate
that unlike what is expected for single-degenerate progenitor systems, the vast
majority of the material accreted by either pure wind mass transfer or a
combination of wind and RLOF mass transfer is helium rather than hydrogen. Even
with extremely optimistic assumptions concerning the mass-transfer and
retention efficiencies, the average mass accreted by systems that eventually
become double-degenerate type Ia SNe is small. Consequently, the lengths of
time that these systems may be SSSs are short, even under optimal conditions,
and the expected number of such systems in a galactic population is negligible.
Conclusions: The population of double-degenerate type Ia SN progenitors that
are SSSs is at least an order of magnitude smaller than the population of
single-degenerate progenitors expected to be SSSs, and the supersoft X-ray
behaviour of double-degenerate systems typically ceases long before the SN
explosions.
