Asymptotic tightness of the Ω(n^{1/4}) untangling bound for planar graphs

Determine whether the known lower bound fix_{𝒢}(n) ∈ Ω(n^{1/4}) for untangling straight-line drawings of planar graphs is asymptotically optimal; equivalently, establish the tight asymptotic order of fix_{𝒢}(n) for the class of all n-vertex planar graphs.

Background

Free sets imply lower bounds on untangling via the relation fix(G) ≥ √k when G has a free set of size k. Current general bounds are Ω(n^{1/4}) and O(n^{0.4965}).

Determining tight asymptotics for fix_{𝒢}(n) would resolve a longstanding gap between lower and upper bounds in geometric untangling of planar graphs.