The fundamental trade-off conjecture for expander-code decoding

Let $G$ be a $(c,d,\alpha,\delta)$-bipartite expander and let $C_0\subseteq\mathbb{F}_2^d$ be an inner code with minimum Hamming distance $d(C_0)\ge d_0$. Let $T(G,C_0)\subseteq\mathbb{F}_2^n$ denote the associated expander code. Expander-code trade-off conjecture. If $\delta d_0>1$, then $T(G,C_0)$ can correct $\Omega_{c,d,\alpha,\delta}(n)$ errors in $O_{c,d,\alpha,\delta}(n)$ time for every such $G$ and $C_0$. This conjecture would make the necessary condition $\delta d_0>1$ essentially sufficient, improving the paper's sufficient condition $\delta d_0>3$ and identifying the fundamental trade-off between expansion and inner-code distance.