Relationship between the edge-connectivity of a mincut graph and its source graph

Determine whether the edge-connectivity of the mincut graph $X(G)$ provides information about, or determines constraints on, the connectivity of the original graph $G$.

Background

The mincut operator encodes intersections among minimum edge-cuts of $G$ as edges of $X(G)$. Although the paper studies convergence and fixed points of this operator, it does not resolve whether graph-connectivity parameters of $X(G)$ can be used to infer corresponding properties of $G$.

This question is posed in the context of determining when mincut graphs remain connected and understanding which source graphs avoid convergence to the null graph.