The test-function convergence conjecture for discrete curvature on LQG maps

Let $\{M_n\}_{n\geq1}$ be a sequence of infinite random planar maps believed to lie in the universality class of $\gamma$-LQG, such as uniform infinite triangulations when $\gamma=\sqrt{8/3}$. Embed each map in the plane using any reasonable embedding, such as the circle-packing or Tutte embedding. Let $c_n$ be an appropriate scaling factor, let $f:\mathbb{C}\to\mathbb{R}$ be a compactly supported $C^2$ test function, and let $\Phi$ be an appropriate variant of the Gaussian free field, with curvature $K_\Phi$ and LQG measure $\mu_\Phi$. The test-function convergence conjecture.

$$\frac{1}{c_n}\sum_{v\in\mathcal{V}M_n}f(v)K_{M_n}(v)\xrightarrow{d}\int_{\mathbb{C}}f(z)K_\Phi(z)\,d\mu_\Phi(z).$$

Moreover, convergence in distribution holds jointly for every finite collection of $C_c^2$ test functions. This gives a precise distributional formulation of the proposed discrete-to-continuum curvature limit, while the appropriate normalization and embedding are left at the level stated in the conjecture.