Connected {2,4}-factor conjecture for 2-tough graphs

Let $G$ be a graph. A connected $\{2,4\}$-factor is a connected spanning subgraph of $G$ in which every vertex has degree either $2$ or $4$. Connected $\{2,4\}$-factor conjecture. Every $2$-tough graph of order at least three admits a connected $\{2,4\}$-factor. This is proposed as a revision of Chvátal's disproved Hamiltonian-cycle conjecture; the source gives no resolution of this stronger factor statement.