Two-size undecidability conjecture for partially fixed-size network coding

A partially fixed-size network has vertex set $V$, edge set $E$, source-message sets $\\{A_v\\}$, demanded-message sets $\\{B_v\\}$, and size specifications $s_M$ and $s_E$. The specifications assign either the default size, represented by $0$, or the fixed size $2$ to every message and edge:

$$s_M(i),s_E(u,v)\in\{0,2\}\quad\text{for all }i,u,v.$$

A network is solvable if there exist encoding and decoding functions satisfying all demands. Two-size undecidability conjecture. The following decision problem is undecidable: given such a partially fixed-size network $(V,E,\{A_v\},\{B_v\},s_M,s_E)$, decide whether it is solvable. The paper has proved undecidability when additional fixed sizes $3$ and $5$ may be used; the conjecture asserts that only fixed size $2$, together with the default size, suffices. The stated obstacle is enforcing the tristate and $(4+1)$-state buffer conditions using only size-$2$ edges.