  It was recently proved in [Corwin-Shen, 2016] that under weak asymmetry
scaling, the height functions for open ASEP on the half-line and on a bounded
interval converge to the Hopf-Cole solution of the KPZ equation with Neumann
boundary conditions. In their assumptions [Corwin-Shen, 2016] chose positive
values for the Neumann boundary conditions, and they assumed initial data which
is close to stationarity. By developing more extensive heat-kernel estimates,
we extend their results to negative values of the Neumann boundary parameters,
and we also show how to generalize their results to narrow-wedge initial data
(which is very far from stationarity). As a corollary via
[Barraquand-Borodin-Corwin-Wheeler, 2017], we obtain the Laplace transform of
the one-point distribution for half-line KPZ, and use this to prove
$t^{1/3}$-scale GOE Tracy-Widom long-time fluctuations.
