  Couplings between topological edge channels open electronic phases possessing
nontrivial eigenmodes far beyond the noninteracting-edge picture. However,
inelastic scatterings mask the eigenmodes' inherent features, often preventing
us from identifying the phases, as is the case for the quintessential
Landau-level filling factor v = 2/3 edge composed of the counter-propagating v
= 1/3 and 1 (1/3-1) channels. Here, we study the coherent-incoherent crossover
of the 1/3-1 channels by tuning the channel length in-situ using a new device
architecture comprising a junction of v = 1/3 and 1 systems, the particle-hole
conjugate of the 2/3 edge. We successfully observed the concurrence of the
fluctuating electrical conductance and the quantized thermal conductance in the
crossover regime, the definitive hallmark of the eigenmodes in the
disorder-dominated edge phase left experimentally unverified.
