On the maximally edge-connected square graphs with degree sequence conditions

Let $G=(V(G),E(G))$ be a finite connected simple graph. The edge-connectivity $λ(G)$ of $G$ is the minimum number of edges that must be removed to make $G$ disconnected. We call $G$ maximally edge-connected when $λ(G)=δ(G)$, where $δ(G)$ is the minimum degree of $G$. The square graph $G^2$ of $G$ is