Equality or square-root growth for higher-genus depth

Determine whether there are infinitely many genera $g$ for which some graph of genus $g$ attains $d(F)=2H(g)-2$, or, alternatively, whether graphs of genus $g$ can attain asymmetric depth $\Theta(\sqrt g)$.

Background

This is the paper’s explicit formulation of the higher-genus tightness problem. It presents two possible resolutions: exact attainment of the Heawood-based bound for infinitely many genera, or a weaker asymptotic construction showing square-root growth in the genus.

References

Is the bound of \Cref{thm:genus-bound} tight? That is, are there infinitely many $g$ for which some graph of genus~$g$ attains $d(F) = 2H(g) - 2$, or even $d(F) = \Theta(\sqrt{g})$?

— Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes  (2609.02585 - Pastorek, 2 Sep 2026) in Question 2, Section 7, “Concluding remarks”