Győri et al.'s planar Turán conjectures for the two particular $\Theta_6$-graphs

Let $ex_{\mathcal{P}}(n,H)$ denote the maximum number of edges in a planar graph with $n$ vertices that does not contain $H$ as a subgraph. For $k\geq 4$, let $\Theta_k$ denote the family of graphs obtained by joining a pair of non-consecutive vertices of a cycle $C_k$ with an edge; write $\Theta^1_6$ and $\Theta^2_6$ for the two particular $\Theta_6$-graphs considered here. Here $O(1)$ denotes a quantity bounded independently of $n$. Győri et al.'s conjecture.

$$ex_{\mathcal{P}}(n,\Theta^1_6)=\frac{45}{17}n+O(1),\qquad ex_{\mathcal{P}}(n,\Theta^2_6)=\frac{18}{7}n+O(1).$$

These conjectures predict the asymptotic planar Turán numbers of the two particular $\Theta_6$-graphs. The paper reports sharp results up to a small additive constant error in one case and infinitely many extremal constructions, while the parser supplies no definitive resolution status for the conjectured equalities.