The toughness condition for connected {2,4}-factors

Let $G$ be a graph of order at least three, and write $\omega(G\setminus S)$ for the number of components of $G\setminus S$. A connected $\\{2,4\\}$-factor is a connected spanning subgraph whose vertex degrees all belong to $\\{2,4\\}$. The connected $\\{2,4\\}$-factor conjecture. If

$$\omega(G\setminus S)\leq \frac{1}{2}|S|+1$$

for every $S\subseteq V(G)$, then $G$ admits a connected $\\{2,4\\}$-factor. The source states that this would simplify the preceding theorem, notes that it is true for strongly $2$-tough graphs, and gives a restricted case in which it can be confirmed.