A cycle plus triangles graph is a 4-regular graph $G$ with a Hamiltonian circuit $C$ and such that the chords of $C$ induce a set of disjoint triangles (3-circuits). A 4-regular graph $G$ has a Hamilton decomposition if its set of edges can be partitioned into two sets that induce Hamiltonian circuits in $G$. A cycle plus triangles graph is edge 3-connected if it doesn’t have any edge 2-cuts (sets of two edges which when removed disconnect the graph).
Our question is the following
Question Let $G$ be a cycle plus triangles graph which is edge 3-connected. Is it true then that $G$ has a Hamiltonian decomposition, or is there a counterexample that might be known?