Conjecture on limit shapes for bead and exponential stochastic interfaces

Let $S=[0,1]^2$, and let $\phi^N$ be a stochastic interface model on $S_N$ with interaction potential $V=\mathsf{bead}$ or $V=\mathsf{\exp}$. Suppose its weight functions are

$$W_{(i,j)}^N(u)=\mathbf{1}_{i=j}\mu u+\mathbf{1}_{i=j=N}Nf_N(u/N),$$

where $f_N(u)\to\infty\mathbf{1}_{u\notin A}$ as $N\to\infty$ and $mu>0$. If the associated partition functions satisfy $N^{-2}\log Z_N\to E\in\mathbb{R}$, let $\bar\phi^N$ be the rescaled interface and let $\sigma^V$ be the surface tension. Limit-shape conjecture. The rescaled interface converges pointwise almost surely to a deterministic function $\xi\in\mathcal{C}^1(S)$, which minimizes

$$\mathcal{E}[v]=\int_S\sigma^V(\nabla v(s,t))\,ds\,dt+\mu\int_0^1v(s,s)\,ds+\infty\mathbf{1}_{v(1,1)\notin A}+E,$$

and $\mathcal{E}[\xi]=0$. This prediction is conditional on the proposed existence of the exponential surface tension and extends the variational description known for more regular interaction potentials; the supplied text does not provide a proof.