Extremal $k$th-eigenvalue conjecture for connected outerplanar graphs

Let $\lambda_k(G)$ be the $k$th largest eigenvalue of the adjacency matrix of a graph $G$. For fixed $k$, let $\lambda_{k,\max}(n)$ be the maximum of $\lambda_k(G)$ over all connected outerplanar graphs on $n$ vertices. Let $P_{q-1}$ be the path on $q-1$ vertices and $K_1\vee P_{q-1}$ its fan graph.

Extremal $k$th-eigenvalue conjecture. If $n=kq+1$, then for fixed $k$ and sufficiently large $n$,

$$\lambda_{k,\max}(n)=\lambda_1(K_1\vee P_{q-1}).$$

Moreover, every extremal graph $G$ on $n$ vertices satisfying $\lambda_k(G)=\lambda_{k,\max}(n)$ has a cut vertex $u$ such that deleting $u$ leaves $k$ copies of $K_1\vee P_{q-1}$.

The paper proves the leading asymptotic behaviour of $\lambda_{k,\max}(n)$, but this exact extremal description remains conjectural.