Bipartite structure conjecture for edge-optimal minimally k-edge-connected graphs

Let $k\geq 3$, and let $G$ be an edge-optimal minimally $k$-edge-connected graph of order $n$, meaning a minimally $k$-edge-connected graph with largest average edge-connectivity among all minimally $k$-edge-connected graphs of order $n$. Edge-bipartite structure conjecture. For $n$ sufficiently large, $G$ is bipartite, with partite sets the set of vertices of degree $k$ and the set of vertices of degree exceeding $k$. The paper establishes sharp bounds and extremal structures for minimally $2$-edge-connected graphs, while the analogous structural problem for $k\geq 3$ remains open.