Bar 1-visibility conjecture for 1-planar graphs

Let $G$ be a graph. It is 1-planar if it has a drawing in the plane in which each edge is crossed by at most one other edge. It is bar 1-visible if it has a bar 1-visibility drawing: vertices are represented by pairwise disjoint horizontal bars, and each edge is represented by a vertical segment joining the bars of its endpoints and crossing at most one bar belonging to another vertex.

Bar 1-visibility conjecture. Every 1-planar graph is a bar 1-visible graph.

The conjecture arises because the authors constructed bar 1-visibility drawings for every 1-planar graph they considered, but did not have a formal proof. The paper gives linear-time constructions for several classes of 1-planar graphs, while the general implication remains open.