  In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions
$G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety
of his examples could be explained by the conjecture that the Dyck path LLT
polynomials exhibit the same phenomenon. In this paper we list the results of
computer explorations which suggest that other examples exhibit the same
phenomenon. We prove two of the resulting conjectures and propose algorithms
that would prove several of our conjectures. In writing this paper we have
learned that similar findings have been independently discovered by Per
Alexandersson.
