Geometric thickness of simple graphs

Determine the computational complexity of the geometric thickness problem for simple graphs, i.e., deciding whether a given simple graph has a straight-line drawing whose edges can be partitioned into k plane layers without crossings within the same layer.

Background

Geometric thickness generalizes simultaneous planarity across multiple layers by requiring straight-line drawings and forbidding same-layer crossings. For multigraphs with sufficiently many layers, ER-completeness has been shown.

The status for simple graphs, including small fixed k, remains unresolved.