  Fix a matroid N. A matroid M is N-fragile if, for each element e of M, at
least one of M\e and M/e has no N-minor. The Bounded Canopy Conjecture is that
all GF(q)-representable matroids M that have an N-minor and are N-fragile have
branch width bounded by a constant depending only on q and N.
  A matroid N stabilizes a class of matroids over a field F if, for every
matroid M in the class with an N-minor, every F-representation of N extends to
at most one F-representation of M.
  We prove that, if Rota's conjecture is false for GF(q), then either the
Bounded Canopy Conjecture is false for GF(q) or there is an infinite chain of
GF(q)-representable matroids, each not stabilized by the previous, each of
which can be extended to an excluded minor.
  Our result implies the previously known result that Rota's conjecture holds
for GF(4), and that the classes of near-regular and sixth-roots-of-unity have a
finite number of excluded minors. However, the bound that we obtain on the size
of such excluded minors is considerably larger than that obtained in previous
proofs. For GF(5) we show that Rota's Conjecture reduces to the Bounded Canopy
Conjecture.
