Planar Turán conjecture for two disjoint 4-cycles

Let $ex_{\mathcal{P}}(n,2C_4)$ denote the maximum number of edges in an $n$-vertex planar graph containing no two vertex-disjoint copies of the cycle $C_4$. Planar Turán conjecture for two disjoint 4-cycles. If $n\geq 23$, then

$$ex_{\mathcal{P}}(n,2C_4)\leq \frac{19}{7}(n-2),$$

and the bound is tight whenever $14\mid(n-2)$. This gives the proposed sharp upper bound for the planar Turán number of two disjoint 4-cycles; the preceding theorem supplies asymptotic lower bounds, while the exact bound and its tightness remain to be established.