Funk–Jackson–Labbate–Sheehan conjecture on 2-factor Hamiltonian bipartite graphs

Let $G$ be a $2$-factor Hamiltonian $k$-regular bipartite graph, meaning that every $2$-factor of $G$ is a Hamiltonian circuit. Let $K_{3,3}$ and $H_0$ be the two specified cubic bipartite graphs, and let a star product be the graph operation used in the construction described in the survey. Funk–Jackson–Labbate–Sheehan's conjecture. Either $k=2$ and $G$ is a circuit, or $k=3$ and $G$ can be obtained from $K_{3,3}$ and $H_0$ by repeated star products. A positive answer would characterize the family of $2$-factor Hamiltonian regular bipartite graphs.