The finite-grid lower-bound conjecture for coded-packet forwarding

Let $p_{k,n,\delta}(\Gamma_m)$ denote the forwarding probability for a grid network of size $m$, and let $Y$ be a binomial random variable with parameters $n$ and $(\theta^+(p))^2$. Fix $\delta \in (0,1/8)$. Finite-grid lower-bound conjecture. For every $k$, $n$, and $m$,

$$p_{k,n,\delta}(\Gamma_m) \ge \inf\{p \mid \Pr(Y \ge k) \ge 1-\delta\}.$$

This formalizes the authors' observation that the large-$m$ approximation appears to provide a lower bound for every finite grid, at least for small $\delta$. The claim is conjectural in the source, and no proof or disproof is supplied.