Linear-size total-variation approximation for the uniform infinite cubic planar graph

Let $(\mathsf{C}_n)_{n\in 2\mathbb{N}}$ be the random cubic planar graph with $V_n$ vertices, and let $\mathcal{D}_i(\mathsf{C}_n)$ and $\mathsf{D}(i)$ denote the associated degree quantities. For a given $\delta$ satisfying

$$0<\delta<\frac{3\kappa}{2},$$

Linear-size approximation conjecture. As $n\in 2\mathbb{N}$ tends to infinity,

$$d_{\mathrm{TV}}\left(\left(\mathcal{D}_i(\mathsf{C}_n)\right)_{1\le i\le \min(\lfloor\delta n\rfloor,V_n)},\left(\mathsf{D}(i)\right)_{1\le i\le\lfloor\delta n\rfloor}\right)\to 0.$$

This conjectures that the independent approximation established for sublinearly many indices remains valid for a positive linear proportion of the graph. The parser gives no resolution evidence, so the status of this approximation remains open.