  We conjecture a combinatorial formula for the monomial expansion of the image
of any Schur function under the Bergeron-Garsia nabla operator. The formula
involves nested labeled Dyck paths weighted by area and a suitable "diagonal
inversion" statistic. Our model includes as special cases many previous
conjectures connecting the nabla operator to quantum lattice paths. The
combinatorics of the inverse Kostka matrix leads to an elementary proof of our
proposed formula when q=1. We also outline a possible approach for proving all
the extant nabla conjectures that reduces everything to the construction of
sign-reversing involutions on explicit collections of signed, weighted objects.
