Classification conjecture for extremal third-eigenvalue graphs

Let $G$ be a graph and let $H_{a,b}$ denote the family of graphs introduced in the paper, where $a,b$ are non-negative integers. Classification conjecture for extremal graphs. If

$$\lambda_3(G) = \frac{|V(G)|}{3} - 1,$$

then $G$ is isomorphic to $H_{a,b}$ for some non-negative integers $a,b$. The conjecture proposes a classification of the graphs attaining the paper's known extremal value for the third largest adjacency eigenvalue; its resolution is not given in the supplied text.