  We study competition of two spreading colors starting from single sources on
the configuration model with i.i.d. degrees following a power-law distribution
with exponent $\tau\in (2,3)$. In this model two colors spread with a fixed and
equal speed on the unweighted random graph.
  We analyse how many vertices the two colors paint eventually. We show that
coexistence sensitively depends on the initial local neighborhoods of the
source vertices: if these neighborhoods are `dissimilar enough', then there is
no coexistence, and the `loser' color paints a polynomial fraction of the
vertices with a random exponent.
  If the local neighborhoods of the starting vertices are `similar enough',
then there is coexistence, i.e., both colors paint a strictly positive
proportion of vertices. We give a quantitative characterization of `similar'
local neighborhoods: two random variables describing the double exponential
growth of local neighborhoods of the source vertices must be within a factor
$\tau-2$ of each other. Both of the two outcomes happen with positive
probability with asymptotic value that is explicitly computable.
  This picture reinforces the common belief that location is an important
feature in advertising.
  This paper is a follow-up of the similarly named paper that handles the case
when the speeds of the two colors are not equal. There, we have shown that the
faster color paints almost all vertices, while the slower color paints only a
random sub-polynomial fraction of the vertices.
