Polynomial-size grid drawing problem for planar graphs

Let $G$ be an arbitrary planar graph. A proper grid drawing of $G$ is a drawing in which vertices are represented by grid points and edges by noncrossing straight-line segments, with no grid point representing a vertex lying in the interior of an edge. Is there a proper grid drawing of $G$ in a grid of polynomial size?

Polynomial-size grid drawing problem. For every planar graph $G$, there is a proper grid drawing of $G$ in a grid of polynomial size.

The paper presents this as an open question because polynomial-size proper grid drawings are known for bounded-degree planar graphs, while the general case remains unresolved.