Faces in girth-saturated graphs on surfaces

What is the maximum length ${\mathrm{f}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\ell },\mathrm{\Sigma })$ of a facial cycle of an inclusion-maximal graph with girth at least $\mathrm{\ell }$ embedded on a given surface $\mathrm{\Sigma }$? If $\mathrm{\Sigma }=\mathcal{P}$ is a plane, we show that $3\mathrm{\ell }-11\le {\mathrm{f}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\ell },\mathcal{P})\le 8\mathrm{\ell }-13$. We also prove that ${\mathrm{f}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\ell },\mathrm{\Sigma })$ is bounded for any integer $\mathrm{\ell }$ and any closed surface $\mathrm{\Sigma }$. For a fixed $\mathrm{\Sigma }$, we show that $\mathrm{\Omega }(\mathrm{\ell })={\mathrm{f}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\ell },\mathrm{\Sigma })=O({\mathrm{\ell }}^2)$, while for a fixed $\mathrm{\ell }\ge 6$, ${\mathrm{f}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\ell },\mathrm{\Sigma })=\mathrm{\Theta }(g)$, where $g$ is the genus of $\mathrm{\Sigma }$.