The NAND 2D regular-grid broadcasting conjecture

Let $P_{X_k}^{+}$ and $P_{X_k}^{-}$ denote the distributions of the observations at layer $k$ conditioned on the source bit being $1$ and $0$, respectively. Fix $\delta\in\big(0,\frac{1}{2}\big)$, and suppose that every vertex with two inputs in the 2D regular grid uses the NAND Boolean rule. NAND broadcasting conjecture. Broadcasting is impossible, in the sense that

$$\lim_{k\rightarrow\infty}\left\|P_{X_k}^{+}-P_{X_k}^{-}\right\|_{\mathsf{TV}}=0.$$

This is a concrete special case of the broader 2D impossibility conjecture. The source presents it as a target for a martingale-based proof, so its resolution is not established in the supplied text.