  The combination of field tunable bandgap, topological edge states, and
valleys in the band structure, makes insulating bilayer graphene a unique
localized system, where the scaling laws of dimensionless conductance g remain
largely unexplored. Here we show that the relative fluctuations in ln g with
the varying chemical potential, in strongly insulating bilayer graphene (BLG)
decay nearly logarithmically for channel length up to L/${\xi}$ ${\approx}$ 20,
where ${\xi}$ is the localization length. This 'marginal' self averaging, and
the corresponding dependence of <ln g> on L, suggest that transport in strongly
gapped BLG occurs along strictly one-dimensional channels, where ${\xi}$
${\approx}$ 0.5${\pm}$0.1 ${\mu}$m was found to be much longer than that
expected from the bulk bandgap. Our experiment reveals a nontrivial
localization mechanism in gapped BLG, governed by transport along robust edge
modes.
