Attainment of the 2-planar crossing-number upper bound

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

Background

The paper notes an analogous upper bound for 2-planar graphs, namely $3.\overline{3}n$ in the cited prior work, and observes that standard drawings of optimal 2-planar graphs achieve the relevant asymptotic behavior. Nevertheless, the crossing number of these graphs is not known, so it remains unresolved whether the bound is attained exactly by either the minimum crossing number over 2-plane drawings or the ordinary crossing number.