We consider the supercritical bisexual Galton–Watson process
(BGWP) with promiscuous mating, that is, a branching process which behaves like
an ordinary supercritical Galton–Watson process (GWP) as long as at
least one male is born in each generation. For a certain example, it was
pointed out by Daley, Hull and Taylor [J. Appl. Probab. 23 (1986)
585–600] that the extinction probability of such a BGWP apparently
behaves like a constant times the respective probability of its asexual
counterpart (where males do not matter) if the number of ancestors grows to
$\infty$. In an earlier paper, we provided general upper and lower bounds for
the ratio between both extinction probabilities and also numerical results that
seemed to confirm the convergence of that ratio. However, theoretical
considerations rather led us to the conjecture that this does not generally
hold. The present article turns this conjecture into a rigorous result. The key
step in our analysis is to identify the extinction probability ratio as a
certain functional of a subcritical ordinary GWP and to prove its continuity as
a function of the number of ancestors in a suitable topology associated with
the entrance Martin boundary of that GWP.