The exponential-profile conjecture for deep extremal level sets

For $h\geq0$, let

$$f_h(\phi):=\sum_{z\in\mathbb Z^2}1_{[0,h]}(\phi_z),$$

where $E_\nu$ denotes expectation under the law $\nu$ used for the local extremal-field profile.

Exponential-profile conjecture. The limit

$$\lim_{h\to\infty}\mathrm e^{-\alpha h}E_\nu(f_h)$$

exists and belongs to $(0,\infty)$.

The source presents this as closely linked to the conjectured asymptotics for deep extremal level-set sizes. No proof or disproof is given.