Uniqueness and KPZ limits for stationary fractional Burgers energy solutions

Consider the fractional stochastic Burgers equation (ec1.5.3). Let $\mathcal{Y}_t^{m,\rho}$ denote a stationary energy solution when it is unique, and let $\mathcal{X}_t^\rho$ be the solution of equation (ec1.4.1). The stationary fractional Burgers conjecture states that there is at most one stationary energy solution of (ec1.5.3), and, writing this unique solution as $\{\mathcal{Y}_t^{m,\rho};t\geq 0\}$, that

$$\lim_{m\to 0}\mathcal{Y}_t^{m,\rho}=\mathcal{X}_t^\rho,$$

while

$$\lim_{m\to +\infty}\mathcal{Y}_{t/m}^{m,1/2}$$

exists and coincides with the KPZ fixed point. The conjecture links uniqueness and the small- and large-nonlinearity limits of stationary energy solutions to the Ornstein–Uhlenbeck regime and the KPZ fixed point; the source does not establish these assertions.