Barraquand and Le~Doussal introduced a family of stationary measures for the
(conjectural) KPZ fixed point on an interval with Neumann boundary conditions,
and predicted that they arise as scaling limit of stationary measures of all
models in the KPZ universality class on an interval. In this paper, we show
that the stationary measures for KPZ fixed point on an interval arise as the
scaling limits of the height increment processes for the open asymmetric simple
exclusion process in the steady state, with parameters changing appropriately
as the size of the system tends to infinity.
