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.