  Recent works of Barraquand and Le Doussal and Bryc, Kuznetsov, Wang, and
Wesolowski gave a description of the open KPZ stationary measure as the sum of
a Brownian motion and a Brownian motion reweighted by a Radon-Nikodym
derivative. Subsequent work of Barraquand and Le Doussal used the Enaud-Derrida
representation of the DEHP algebra to formulate the open ASEP stationary
measure in terms of the sum of a random walk and a random walk reweighted by a
Radon-Nikodym derivative. They show that this Radon-Nikodym derivative
converges pointwise to the Radon-Nikodym derivative that characterizes the open
KPZ stationary measure. This article proves that the corresponding sequence of
measures converges weakly to the open KPZ stationary measure. This provides an
alternative proof of the probabilistic formulation of the open KPZ stationary
measure, which avoids dealing explicitly with finite dimensional distributions.
We also provide the first construction of the measure on intervals of a general
length and for the full range of parameters in the fan region $(u+v>0)$.
