Irreducible quantum-isomorphic pairs below 3-connectivity

Construct an irreducible pair of quantum-isomorphic graphs that are not isomorphic, with at least one graph not 3-connected.

Background

An irreducible pair is defined as a pair of quantum-isomorphic, non-isomorphic graphs for which quantum-isomorphic induced subgraphs are necessarily isomorphic. The paper proves that both graphs in any such pair must be 2-connected. This question asks whether that lower bound can be improved to 3-connectedness, or equivalently whether an irreducible example can fail to be 3-connected.