The corrected-limit conjecture for multidimensional stochastic Burgers nonlinearities

Let $u=(u_1,\ldots,u_d)$ be an $\mathbb{R}^d$-valued process, let $h:\mathbb{R}^d\to\mathbb{R}^d$ be smooth with bounded second and third derivatives, and let $u^{\varepsilon}$ solve the approximating system

$$du_i^{\varepsilon}=\nu\,\partial_x^2u_i^{\varepsilon}\,dt+\sum_{j=1}^d\partial_jh_i(u^{\varepsilon})D_{\varepsilon}u_j^{\varepsilon}\,dt+\sigma\,dw_i(t).$$

Multidimensional correction conjecture. As $\varepsilon\to0$, $u^{\varepsilon}$ converges to the solution of

$$du_i=\nu\,\partial_x^2u_i\,dt+\sum_j\left(\partial_jh_i(u)\,\partial_xu_j-\frac{\sigma^2}{4\nu}\partial_{jj}^2h_i(u)\right)dt+\sigma\,dw_i.$$

The predicted correction comes from the diagonal high-frequency contributions, under the cutoff condition $N\ll\varepsilon^{-1}$. The source states this as an expected behaviour and gives no proof.