  The Hanna Neumann conjecture gives a bound on the intersection of finitely
generated subgroups of free groups. We explore a natural extension of this
result, which turns out to be true only in the finite index case, and provide
counterexamples for the general case. We also see that the graph-based method
of generating random subgroups of free groups developed by Bassino, Nicaud and
Weil is well-suited to generating subgroups with non-trivial intersections. The
same method is then used to generate a counterexample to a similar conjecture
of Guzman.
