Commutation of deformation and scaling limits for the Laplace-Beltrami operator

Let $\Omega_1$ and $\Omega_2$ be the regions supporting deformations with parameters $\epsilon_1$ and $\epsilon_2$, let ${\text{\IceMountain}}$ be the Laplace-Beltrami operator on the deformed triangulation, and let ${\text{\IceMountain}}_{\mathrm{cr}}$ be the operator for the critical triangulation $T_{\mathrm{cr}}$. Let $\delta_1{\text{\IceMountain}}$ and $\delta_2{\text{\IceMountain}}$ denote the corresponding first variations, and let $F_1,F_2$ be the associated deformation fields.

Commutation-of-limits conjecture. For sufficiently small $\epsilon_1$ and $\epsilon_2$, the scaling limit exists uniformly with respect to $T_{\mathrm{cr}}$; the deformation limits are uniform and commute with the scaling limit, with

$$\lim_{\substack{\epsilon_1\to0\epsilon_2\to0}}\lim_{\ell\to\infty}\frac{1}{\epsilon_1\epsilon_2}\operatorname{tr}\left[\delta_1{\text{\IceMountain}}\cdot{\text{\IceMountain}}_{\mathrm{cr}}^{-1}\cdot\delta_2{\text{\IceMountain}}\cdot{\text{\IceMountain}}_{\mathrm{cr}}^{-1}\right] =\lim_{\ell\to\infty}\lim_{\substack{\epsilon_1\to0\epsilon_2\to0}}\frac{1}{\epsilon_1\epsilon_2}\operatorname{tr}\left[\delta_1{\text{\IceMountain}}\cdot{\text{\IceMountain}}_{\mathrm{cr}}^{-1}\cdot\delta_2{\text{\IceMountain}}\cdot{\text{\IceMountain}}_{\mathrm{cr}}^{-1}\right] =\iint_{\Omega_1\times\Omega_2}dx_1\,dx_2\,\frac{\bar\partial F_1(x_1)\bar\partial F_2(x_2)}{(x_1-x_2)^4}+\text{c.c.}$$

The conjecture asserts that the perturbative and thermodynamic/scaling limits can be interchanged without geometric restrictions on the infinite critical triangulation. The paper presents it as an unresolved claim motivated by uniform estimates.