The conjectural two-dimensional Gaussian free field fluctuation theorem for random height functions

Assume the notation of Theorem TCLT. Let $\mathcal{H}(q^{-t},v)$ be the random height function, let $\mathbb{P}_\varepsilon$ denote its law, and let $\Omega$ be the map to the upper half-plane associated with the ellipse $\tilde{Q}(u,v)=0$. For each $u$, let $a(u)$ and $b(u)$ be the $v$-coordinates where the vertical line through $u$ intersects this ellipse. For polynomials $f_i\in\mathbb{R}[x]$, numbers $u_i\in(1,q^{-T})$, and sequences $t_i(\varepsilon)$ satisfying $q^{-t_i(\varepsilon)}=u_i+O(\varepsilon)$, define the centered averages

$$\int_{\mathbb{R}}\sqrt{\pi}\left(\mathcal{H}(q^{-t_i},v)-\mathbb{E}_{\mathbb{P}_\varepsilon}[\mathcal{H}(q^{-t_i},v)]\right)f_i(v)\,dv.$$

Conjectural two-dimensional fluctuation theorem. The centered random height function, multiplied by $\sqrt{\pi}$, converges in distribution to the pullback under $\Omega$ of the Gaussian free field with Dirichlet boundary conditions on the upper half-plane $\mathbb{H}$. More precisely, the preceding averages converge jointly to the corresponding GFF averages

$$\int_{a(u_i)}^{b(u_i)}\mathcal{F}(\Omega(u_i,y))f_i(y)\,dy,$$

which form a centered Gaussian vector $(X_1,\ldots,X_m)$ with covariance

$$\mathbb{E}[X_iX_j]=\int_{a(u_i)}^{b(u_i)}\int_{a(u_j)}^{b(u_j)}f_i(x)f_j(y)\left(-\frac{1}{2\pi}\log\left|\frac{\Omega(u_i,x)-\Omega(u_j,y)}{\Omega(u_i,x)-\overline{\Omega}(u_j,y)}\right|\right)\,dx\,dy.$$

This is presented as a conjectural two-dimensional extension of the paper's one-dimensional fluctuation theorem. It predicts Gaussian free field fluctuations for the height function after the appropriate macroscopic scaling and conformal pullback; the statement itself does not provide a proof or resolution.