The maximum-spread outerplanar graph conjecture

Let $P_k$ denote the path on $k$ vertices, let $K_1$ denote the one-vertex graph, and let $G\vee H$ denote the join of graphs $G$ and $H$. The spread of a graph is the difference between the largest and most negative eigenvalue of its adjacency matrix. An outerplanar graph is a graph that can be drawn in the plane without crossings with every vertex incident with the unbounded face.

Maximum-spread outerplanar graph conjecture. For $n$ sufficiently large, the unique $n$-vertex outerplanar graph of maximum spread is

$$K_1\vee P_{n-1}.$$

The paper's main theorem shows that every sufficiently large extremal outerplanar graph has the form $K_1\vee F$, where $F$ is a linear forest with $\Omega(n)$ edges. The conjecture identifies the remaining forest as the path $P_{n-1}$ and asserts uniqueness; the source leaves this determination open.