Cohen–Macaulayness under contracting the connecting edge

Determine whether the Cohen–Macaulay property of the edge ring K[G] is preserved and reflected by contracting the connecting edge e = {x,y} in graphs formed by joining two disjoint connected graphs G1 and G2 by e; specifically, decide whether K[G] is Cohen–Macaulay if and only if K[G e] is Cohen–Macaulay.

Background

The paper establishes Cohen–Macaulay preservation for certain even simple path contractions under additional path containment hypotheses (Theorem 2.9 and Corollary 2.10). In contrast, for edge contractions in graphs connected by e, only partial results are available: β1 invariance (Theorem 4.2) and full Betti preservation in the bipartite component case (Proposition 4.3). Examples indicate subtle dependence on the connection vertices.

This question asks whether Cohen–Macaulayness is invariant and bi-implicative under contracting the connecting edge across the entire class of graphs connected by e.