Ghosh–Győri–Paulos–Xiao conjecture on the planar Turán number of the balanced double star S_{3,3}

A balanced double star $S_{m,m}$ is the graph obtained from an edge whose endpoints are joined respectively to $m$ and $m$ additional distinct vertices. For a planar graph family, let $\mathrm{ex}_{\mathcal{P}}(n,S_{3,3})$ denote the maximum number of edges in an $n$-vertex planar graph containing no subgraph isomorphic to $S_{3,3}$. Ghosh–Győri–Paulos–Xiao's conjecture.

$$\mathrm{ex}_{\mathcal{P}}(n,S_{3,3}) = \begin{cases} 3n-6 & \text{if } 3\leq n\leq 7,\\ 16 & \text{if } n=8,\\ 18 & \text{if } n=9,\\ \lfloor 5n/2\rfloor-5 & \text{otherwise}. \end{cases}$$

This conjecture would determine the planar Turán number of the balanced double star $S_{3,3}$ for every $n$. The source reports only the bounds $\lfloor5n/2\rfloor-5\leq \mathrm{ex}_{\mathcal{P}}(n,S_{3,3})\leq \lfloor5n/2\rfloor-2$ for $n\geq 3$, so the conjecture remains open in the supplied context.