Discrete Gaussian free field ALE convergence under exponential observation

Let $\phi_n$ be a zero-boundary discrete Gaussian free field converging to a continuum Gaussian free field $\Phi$, and let $\mathbb{A}_{-\lambda,\lambda}$ denote the continuum ALE. For sufficiently small $T$, form the discrete ALE associated with the imaginary part of $\exp(iT\phi_n)$. Exponential-observation ALE conjecture. This discrete ALE converges to $\mathbb{A}_{-\lambda,\lambda}$.

The conjecture asks whether the natural local ALE construction from the exponential observation agrees with the deterministic reconstruction limit already obtained in the paper. The supplied text gives no resolution.