  The $\nu = 2/3$ filling is the simplest paradigmatic example of a fractional
quantum Hall state, which contains counter-propagating edge modes. These modes
can be either in the unequilibrated regime or equilibrated to different
extents, on top of the possible edge reconstruction. In the unequilibrated
regime, two distinct renormalization group fixed points have been previously
proposed, namely Kane-Fischer-Polchinski and Wang-Meir-Gefen. In the
equilibration regime, different degree of thermal equilibration can exist,
while charge is fully equilibrated. Here, we show that these rich variety of
models can give rise to three possible conductance plateaus at $e^2/2h$
(recently observed in experiments), $5e^2/9h$ (predicted here), and $e^2/3h$
(observed earlier in experiments) in a quantum point contact geometry. We
identify different mechanisms for \emph{electrical shot noise} generation at
these plateaus, which provides an experimentally accessible venue for
distinguishing among the distinct models.
