Undecidability of the asymptotic capacity threshold for partially fixed-size networks

For a partially fixed-size network, let $k_1$ be the default alphabet size for default-size messages and $k_2$ the default alphabet size for default-size edges. Let $k_1^*(k_2)$ be the largest possible $k_1$ for fixed $k_2$ such that the network admits a coding scheme, and define its asymptotic capacity by

$$\underset{k_2\to\infty}{\limsup}\frac{\log k_1^*(k_2)}{\log k_2}.$$

Asymptotic-capacity undecidability conjecture. The following problem is undecidable: given a partially fixed-size network $(V,E,\{A_v\},\{B_v\},s_M,s_E)$, decide whether its asymptotic capacity is at least $1$. This is proposed as a future direction; the source gives no resolution or supporting reduction beyond stating the conjecture.