Tight crossing number for 3-planar graphs

Determine whether there exist 3-planar graphs on n vertices whose 3-planar crossing number or ordinary crossing number attains the upper bound 5.5(n-2).

Background

The paper proves that every 3-plane drawing of a graph on n vertices has at most 5.5(n-2) crossings, and consequently that every 3-planar graph G satisfies cr(G) ≤ cr₃(G) ≤ 5.5(n-2). The bound is tight for 3-plane drawings, but tightness for the minimum crossing numbers cr₃(G) and cr(G) of individual 3-planar graphs is not established. The authors therefore ask whether either crossing-number parameter can attain this bound.