Exclusion of infinite-specific-free-energy gradient Gibbs measures

Let $U_{\phi}$ be the set appearing in the source, and let $\mu$ be a non-trivial gradient Gibbs measure, meaning that $\mu$-almost surely $Z_{\Lambda}(\phi)<\infty$ for every finite $\Lambda\subset\subset\mathbb Z^d$. Let $S(\mu)$ denote its slope and $SFE(\mu)$ its specific free energy. Finite-specific-free-energy conjecture. For every $u\in U_{\phi}$ and every perturbed simply attractive potential, there exists no such measure $\mu$ with

$$S(\mu)=u\quad\text{and}\quad SFE(\mu)=\infty.$$

The source presents this as a refinement of its variational-principle results, which identify specific free energy with surface tension when the former is finite; the asserted exclusion of the infinite case remains open.