Conjecture on existence of surface tension for the exponential interaction

Let \exp} denote the exponential interaction, and for each $N$ let $\sigma_N^{\mathsf{\exp}}$ be its finite surface tension. Exponential surface-tension conjecture. The limit

$$\sigma^{\mathsf{\exp}}:=\lim_{N\to\infty}\sigma_N^{\mathsf{\exp}}$$

exists. Surface tension is intended to describe the asymptotic energy cost of an interface with a prescribed tilt; existence would permit variational analysis of the exponential model, whose interaction does not satisfy the hypotheses of the cited Funaki--Spohn result.