The finite-difference correction conjecture for stochastic Burgers equation

Let $u^N$ be the solution of the finite-difference approximation with an even number $N$ of modes, and let $\nu>0$ and $\sigma$ denote the viscosity and noise parameters. Finite-difference correction conjecture. As $N\to\infty$, $u^N$ converges to the solution of

$$du=\nu\,\partial_x^2u\,dt-u\,\partial_xu\,dt+\frac{\sigma^2}{4\nu}\,dt+\sigma\,dw(t).$$

The predicted constant drift is obtained from the zero mode of the discrete nonlinear term. The source gives this as an expectation based on the Fourier calculation, with no proof or resolution stated.