  The planar Turan number $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ is the
largest number of edges in an $n$-vertex planar graph with no $\ell$-cycle. For
$\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$
are known that hold with equality infinitely often. Ghosh, Gy\"{o}ri, Martin,
Paulo, and Xiao [arxiv:2004.14094] conjectured an upper bound on
$\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ for every $\ell\ge 7$ and $n$
sufficiently large. We disprove this conjecture for every $\ell\ge 11$. We also
propose two revised versions of the conjecture.
