Scaling-limit conjecture for the wetting-model dynamics

Let $K=\{h\in H:h\geq 0\ \text{a.e.}\}$, where $H=L^2(0,1)$, and let $\mu$ be the law on $K$ of a reflecting Brownian motion on $[0,1]$. Let $(u_t)_{t\geq 0}$ be the reversible Markov process associated with the Dirichlet form

$$\mathcal{E}(f,g)=\frac12\int\langle\nabla f,\nabla g\rangle\,\mathrm{d}\mu,$$

started from equilibrium, so that $u_0\overset{(d)}{=}\mu$. Let $(Y_t^N)_{t\in[0,T]}$ denote the rescaled wetting-model process. Scaling-limit conjecture. For all $T>0$, as $N\to\infty$, $(Y_t^N)_{t\in[0,T]}$ converges in law in $C([0,T],H^{-1}(0,1))$ to $(u_t)_{t\in[0,T]}$. The conjecture identifies the dynamical scaling limit with the Markov process whose formal equation includes reflection at zero; the source establishes tightness of the approximating family but leaves identification of the limit conjectural.