The linear edge bound conjecture for graphs with at most two bends

Let $R_2$ be the class of graphs that can be drawn in the plane with vertices as points and edges as polygonal arcs having at most two bends, such that any two edge-arcs cross at right angles and not at a bend. For a graph $G$ on $n$ vertices, membership in $R_2$ means $G$ has such a drawing.

Linear edge bound conjecture. A graph $G$ on $n$ vertices belonging to the class $R_2$ can have at most $O(n)$ edges.

The paper proves the weaker bound $O(n\log^2 n)$ for graphs in $R_2$; the conjectured linear bound would further narrow the gap between the classes $R_2$ and $R_3$.