  The Nielsen number $N(f)$ is a lower bound for the minimal number of fixed
points among maps homotopic to $f$. When these numbers are equal, the map is
called Wecken. A recent paper by Brimley, Griisser, Miller, and the second
author investigates the abundance of Wecken maps on surfaces with boundary, and
shows that the set of Wecken maps has nonzero asymptotic density.
  We extend the previous results as follows: When the fundamental group is free
with rank $n$, we give a lower bound on the density of the Wecken maps which
depends on $n$. This lower bound improves on the bounds given in the previous
paper, and approaches 1 as $n$ increases. Thus the proportion of Wecken maps
approaches 1 for large $n$. In this sense (for large $n$) the known examples of
non-Wecken maps represent exceptional, rather than typical, behavior for maps
on surfaces with boundary.
