Monotonicity conjecture for the second partition eigenvalue

Let $\mathcal{G}$ be a finite, compact, connected metric graph. An exhaustive rigid $2$-partition of $\mathcal{G}$ is a partition ${\mathcal P}_2=\{\mathcal{G}_1,\mathcal{G}_2\}$ whose two parts are exhaustive and rigid. The quantity $\mu_2(\mathcal{G})$ denotes the relevant second eigenvalue of the graph. Monotonicity conjecture. There exists an exhaustive rigid $2$-partition ${\mathcal P}_2=\{\mathcal{G}_1,\mathcal{G}_2\}$ such that

$$\mu_2(\mathcal{G})\leq\min\{\mu_2(\mathcal{G}_1),\mu_2(\mathcal{G}_2)\}.$$

This is a necessary condition for the expected monotonicity property of the optimal partition energy as the number of parts increases; the paper records it as a seemingly obvious conjecture and indicates that it is used later in the discussion of extremal partitions.