Fujita–Nakamigawa conjecture on the balanced decomposition number

Let $G$ be a connected graph. Its balanced decomposition number is the minimum integer $s$ such that, for every balanced coloring $V(G)=P_1\uplus P_2\uplus X$ with $|P_1|=|P_2|$, there is a partition $V(G)=V_1\uplus\cdots\uplus V_r$ in which every induced subgraph $G[V_i]$ is connected, $|V_i\cap P_1|=|V_i\cap P_2|$, and $|V_i|\leq s$. A graph is 2-connected if it has vertex-connectivity at least $2$.

Fujita–Nakamigawa's conjecture. If $G$ is $2$-connected, then its balanced decomposition number is at most

$$\left\lfloor\frac{|V(G)|}{2}\right\rfloor+1.$$

This conjecture connects the balanced decomposition number with vertex-connectivity and was motivated by pebble motion on graphs. It was announced as solved by G. J. Chang and N. Narayanan.