Hudák–Madaras–Suzuki conjecture on 3-connected maximal 1-planar graph size

Let $\overline{\mathcal{P}^\star}$ be the family of 3-connected maximal 1-planar graphs, and let $m(\overline{\mathcal{P}^\star},n)$ denote the minimum number of edges among graphs in this family with $n$ vertices. Hudák–Madaras–Suzuki conjecture.

$$m(\overline{\mathcal{P}^\star},n)=\frac{18}{5}n+c,$$

where $c$ is a constant. The paper states that its results disprove the related conjecture that every 3-connected maximal 1-planar graph has at least $\frac{18}{5}n+O(1)$ edges.