The minimum-energy conjecture for normal cycle sets

Let $G$) be a plane digraph of digirth $g$, and let $\mathcal{N}$ be a normal set of cycles of $G$ of size $fvs(G)$ minimizing $q(\mathcal{N})=m_\mathcal{N}-n_\mathcal{N}$. Here $m_\mathcal{N}$ and $n_\mathcal{N}$ denote the associated edge and vertex counts, and $\mathbf{E_{tot}}(\mathcal{N})$ is the total energy of the normal set.

The minimum-energy conjecture.

$$\mathbf{E_{tot}}(\mathcal{N})\geq \frac{2}{g}(m_\mathcal{N}-n_\mathcal{N}).$$

This conjecture would provide the energy lower bound needed to improve the general upper bound on the minimum feedback vertex set for planar digraphs of fixed digirth. Its status is not resolved in the source.