Invariant measures for the open KPZ equation: the Gaussian case

In [arXiv:2409.08465], Quastel and Gu use Stein's equation and integration by parts to give a direct proof that drifted Brownian motions are stationary (modulo height shifts) for the full-line KPZ equation. In this article, we consider the open KPZ equation with boundary conditions $\partial_x h(t,0) = \partial_x h(t,1) = \alpha$ for a general real parameter $\alpha$, and emulate the approach of Quastel and Gu to provide a similar proof that Brownian motion with constant drift $\alpha$ is invariant (modulo height shifts) in this case.