Tight crossing number for 2-planar graphs

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

Background

The paper notes that recent work gives the upper bound cr(G) ≤ cr₂(G) ≤ 3.\overline{3}n for 2-planar graphs, while the standard drawings of optimal 2-planar graphs provide evidence that a bound of 3.\overline{3}(n-2) may be tight up to an additive constant. However, the crossing numbers of those graphs are not known. The authors explicitly ask whether either the minimum crossing number over 2-plane drawings, cr₂(G), or the ordinary crossing number, cr(G), can equal 3.\overline{3}(n-2).