Parametric optimal-partition conjecture for metric graphs

Let $\mathcal{G}$ be a metric graph and let $k\geq1$. A rigid $k$-partition is a partition of $\mathcal{G}$ of the type considered in the paper, and $\mathcal{P}$ is called internally connected when each of its partition subgraphs is connected. Write $\mathcal{D}_p(\mathcal{P})$ for its $p$-energy and $\mathcal{D}_{k,p}^{\mathrm{opt}}(\mathcal{G})$ for the optimal value. Parametric optimal-partition conjecture.

1. If there exists an exhaustive rigid $k$-partition $\mathcal{P}$ achieving $\mathcal{D}_{k,\infty}^{\mathrm{opt}}(\mathcal{G})$ which is not internally connected, then $\mathcal{P}$ does not achieve $\mathcal{D}_{k,p}^{\mathrm{opt}}(\mathcal{G})$ for any $p<\infty$; equivalently,

$$\mathcal{D}_p(\mathcal{P})>\mathcal{D}_{k,p}^{\mathrm{opt}}(\mathcal{G})$$

for all $p<\infty$.

2. Whenever there exists a rigid $k$-partition $\mathcal{P}$ achieving $\mathcal{D}_{k,\infty}^{\mathrm{opt}}(\mathcal{G})$ but failing to achieve $\mathcal{D}_{k,p}^{\mathrm{opt}}(\mathcal{G})$ for some $p<\infty$, the function $p\mapsto\mathcal{D}_{k,p}^{\mathrm{opt}}(\mathcal{G})$ is strictly monotonic.

The conjecture describes how optimal partitions should vary with $p$ when an optimal infinity-partition is not internally connected. The paper notes that this behaviour is supported by the equilateral three-star example, while the general assertions are left open.