Sequential zero-width limit conjecture for the sticky wetting dynamics

For $\eta,\epsilon>0$, let $(u_t^{\eta,\epsilon})_{t\geq 0}$ be the stationary $K$-valued Markov process associated with the Dirichlet form $\mathcal{E}^{1,\eta,\epsilon}$ and started from equilibrium, with $u_0^{\eta,\epsilon}\overset{(d)}{=}P_0^{1,\eta,\epsilon}$. Here $H^{-1}(0,1)$ is the completion of $H=L^2(0,1)$ under

$$\|f\|_{-1}^2=\sum_{n=1}^{\infty}n^{-2}|\langle f,e_n\rangle|^2,$$

where $e_n(\theta)=\sqrt{2}\sin(n\pi\theta)$ for $\theta\in[0,1]$. Let $(u_t)_{t\geq 0}$ be the Markov process from the scaling-limit conjecture. Sequential zero-width limit conjecture. For all $T>0$, $(u_t^{\eta,\epsilon})_{t\in[0,T]}$ converges weakly to $(u_t)_{t\in[0,T]}$ in $C([0,T],H^{-1}(0,1))$ as $\epsilon$ tends to zero and then $\eta$ tends to zero. This conjecture proposes convergence of the regularized attractive mechanism at $\eta$ to the limiting reflected process; the source gives no resolution status beyond stating the conjecture.