  We study positivity notions for the tropicalization of type A flag varieties
and the flag Dressian. We focus on the hollow case, where we have one
constituent of rank 1 and another of corank 1. We characterize the three
different notions that exist for this case in terms of Pl\"ucker coordinates.
These notions are the tropicalization of the totally non-negative flag variety,
Bruhat subdivisions and the non-negative flag Dressian. We show the first
example where the first two concepts above do not coincide. In this case the
non-negative flag Dressian equals the tropicalization of the Pl\"ucker
non-negative flag variety and is equivalent to flag positroid subdivisions.
Using these characterizations, we provide similar conditions on the Pl\"ucker
coordinates for flag varieties of arbitrary rank that are necessary for each of
these positivity notions, and conjecture them to be sufficient. In particular
we show that regular flag positroid subdivisions always come from the
non-negative Dressian. Along the way, we show that the set of Bruhat interval
polytopes equals the set of twisted Bruhat interval polytopes and we introduce
valuated flag gammoids.
