The complete-bipartite extremal conjecture for algebraic connectivity

Let $G$ be a graph with exactly $n$ vertices and $m=2(n-2)$ edges. The complete-bipartite extremal conjecture. Among all such graphs, a graph maximizing the algebraic connectivity is the complete bipartite graph $K_{2,n-2}$, and

$$\lambda_{2}(K_{2,n-2})=2.$$

This conjecture concerns the graph with four fewer edges than the complete bipartite graph parameterized by its smaller part. The source presents it as an open conjecture motivated by numerical and extremal questions about maximizing algebraic connectivity.