Minimum-size conjecture for 3-connected graphs with cyclic neighborhoods

Let $G$ be a graph of order $n$ and size $m$. Assume that $G$ is 3-connected and that every neighborhood of a vertex contains a cycle. Minimum-size conjecture. Then

$$m\geq \frac{7}{3}(n-1).$$

The question concerns graphs with no forest cut: such graphs are necessarily 3-connected and have a cycle in every vertex neighborhood. The family $G_k$ described in the paper has $3k+4$ vertices and $7k+7$ edges and satisfies these structural conditions, motivating the proposed lower bound; whether any such graph can have fewer edges remains open.