  We prove results for first-passage percolation on the configuration model
with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights
with strictly positive support of the form Y=a+X, where a is a positive
constant. We prove that the weight of the optimal path has tight fluctuations
around the asymptotical mean of the graph-distance if and only if the following
condition holds: the random variable X is such that the continuous-time
branching process describing first-passage percolation exploration in the same
graph with excess edge weight X has a positive probability to reach infinitely
many individuals in a finite time. This shows that almost shortest paths in the
graph-distance proliferate, in the sense that there are even ones having tight
total excess edge weight for various edge-weight distributions.
