It is a notorious open problem to find a smallest set of $N$ points that
permit any $n$-vertex planar graph to be drawn in the plane without
crossings, using only those $N$ points as vertices, and straight segments
as edges. An upperbound of $O(n^2)$
is provided by integer grids, but the best lowerbound is linear in $n$.
A point set that allows such plane drawings for every $n$-vertex planar
graph is called a *universal set*.

My question is:

. Are there subquadratic universal sets for drawing $1$-plane graphs, graphs that have a straight-line drawing where each edge is crossed by at most one other edge? If this is not known, are there subquadratic universal sets for drawing $k$-plane graphs, for some $k > 1$?Q

As $k$ approaches $n$, there should be linear-size universal sets.

^{ Fig.1: Chaplick, Steven, Fabian Lipp, Alexander Wolff, and Johannes Zink. }

^{ "Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends." arXiv:1806.10044 (2018). }