Let $G=(V,E)$ be an undirected graph with $|V|\geq 4$ such that for any distinct vertices $a_1,a_2,b_1,b_2$, there is a path from $a_1$ to $a_2$ and a (vertex-)disjoint path from $b_1$ to $b_2$ (in other words, the graph is $2$-linked).

Assign a nonnegative real number to each vertex in $V$. Suppose the sum of these numbers is $2s$, and a subset of them has sum exactly $s$. What is the largest constant $c$ for which (regardless of $G$ and $s$) there always exists a subset $V'\subseteq V$ such that both $V'$ and $V\backslash V'$ form connected subgraphs, and the sum of the numbers in $V'$ belongs to $[cs,s]$?

The following example shows that $c\leq 5/6$: a clique $K_5$ with one edge $e$ removed, numbers $3,3$ on the two vertices adjacent to $e$, and $2,2,2$ on the remaining vertices. Another example is to take $K_{3,3}$ with $3,3,3$ on one side and $1,1,1$ on the other. Is $c=5/6$ tight?

(It is easy to see that such a constant $c$ must exist: $c=0$ works since for any connected graph, its vertices can be partitioned into two parts such that both parts are connected.)