The superharmonic-function conjecture for the NAND 2D regular grid

Let $\mathcal{Y}^*=\bigcup_{k\in\mathbb{N}\setminus\{0\}}\mathcal{Y}^k$ be the set of nonempty finite strings over $\mathcal{Y}$, let $\{Y_k:k\in\mathbb{N}\}$ be the Markov chain associated with the NAND 2D regular grid, and let $\{\mathcal{F}_k:k\in\mathbb{N}\}$ be the filtration generated by the chain and the binary symmetric channels before level $k$. Superharmonic-function conjecture. For every $\delta\in\big(0,\frac{1}{2}\big)$, there exists a Borel-measurable superharmonic function $f_\delta:\mathcal{Y}^*\to\mathbb{R}$ such that $\{f_\delta(Y_k):k\in\mathbb{N}\}$ is an $\{\mathcal{F}_k\}$-adapted supermartingale and, for some constant $C=C(\delta)>0$,

$$\mathbb{E}\!\left[f_\delta(Y_{k+1})\middle|\mathcal{F}_k\right]=\mathbb{E}\!\left[f_\delta(Y_{k+1})\middle|Y_k\right]\leq f_\delta(Y_k),$$

with $f_\delta(Y_k)\geq C N_k$ almost surely for every $k\in\mathbb{N}$. Such a family would provide the supermartingale needed for the proposed impossibility proof, but existence is conjectural in the supplied text.