The low-density normal-set conjecture

A plane digraph is considered with its directed cycles and normal sets. A normal set of cycles $\mathcal{N}$ has low density when

$$2m_\mathcal{N}\leq 3n_\mathcal{N},$$

where $m_\mathcal{N}$ and $n_\mathcal{N}$ are the associated edge and vertex counts; equivalently, $q(\mathcal{N})\leq m_\mathcal{N}/3$.

Low-density normal-set conjecture. Every plane digraph $G$ admits a normal set of cycles with low density and of size $fvs(G)$.

If true, this would imply the upper bound $fvs(G)\leq 3n/(2g)$ for planar digraphs of digirth $g$, improving the general bound for small $g$.