Quadratic-logarithmic weak coloring numbers for planar graphs

Prove or refute that the weak r-coloring number of every planar graph G is O(r^2 log r).

Background

The survey records an O(r3) upper bound for planar graphs and an O(r2 log r) bound for planar graphs of bounded treewidth. It identifies the extension of the latter bound to all planar graphs as a principal open problem.

References

This bound is tight for planar graphs of bounded treewidth. It is one of the main open problems whether in general the weak coloring numbers of planar graphs is bounded by $O(r2\cdot \log r)$.

\begin{problem} Is it true that $\wcol_r(G)\in O(r2\log r)$ for every planar graph $G$? \end{problem}

— On the generalized coloring numbers  (2501.08698 - Siebertz, 15 Jan 2025) in Problem, Section 4.3, 'Planar graph and graphs of bounded genus'