The linkedness characterization of the two-agent price of connectivity

Let $G$ be a biconnected graph. For positive integers $a,b$, say that $G$ is $(a,b)$-linked if, for every pair of disjoint vertex sets $M_1,M_2$ with $|M_1|=a$ and $|M_2|=b$, there are disjoint connected subgraphs $G_1,G_2$ such that $M_i\subseteq G_i$ for $i=1,2$. Let $\operatorname{PoC}(G,2)$ denote the price of connectivity for two agents. Linkedness conjecture. If $k\geq 2$ is an integer and $G$ is $(2,k-1)$-linked but not $(2,k)$-linked, then

$$\operatorname{PoC}(G,2)=\frac{2k-1}{2k}.$$

The conjecture would determine the price of connectivity for the two-agent case in terms of linkedness; the paper verifies it for complete graphs with an arbitrary matching removed, but does not establish it for general biconnected graphs.