The correction-limit conjecture for asymmetric Burgers discretisations

Let $u$ solve the stochastic Burgers equation with viscosity $u$, noise amplitude $\sigma$, and asymmetric finite-difference parameters $a,b$. Let $u^{\varepsilon}$ denote the solution of the corresponding approximating equation. Correction-limit conjecture. The solution $u^{\varepsilon}$ converges as $\varepsilon\to0$ to the solution of

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

Thus, the approximation converges to the stochastic Burgers equation only when $a=b$. This prediction is based on the non-vanishing zero-mode correction produced by the asymmetric discretisation; the source presents it as an expectation rather than a proved result.