Attainment of the 3-planar crossing-number upper bound

Determine whether there exist 3-planar graphs on n vertices whose 3-planar crossing number equals 5.5(n-2), or whose ordinary crossing number equals 5.5(n-2).

Background

The paper proves that every 3-plane drawing of a graph on n vertices contains at most 5.5(n-2) crossings, and that this bound is tight for 3-plane drawings in the sense that there are drawings attaining it. However, tightness of the universal drawing bound does not establish that some graph has minimum 3-plane crossing number or ordinary crossing number equal to this value. The authors therefore explicitly ask whether either form of equality is attained by a 3-planar graph.