This question is related to the question of drawing a combinatorial 3-configuration of points and lines with straight lines. We only relax the condition and admit drawings with pseudolines. Let us call a combinatorial configuration that can be drawn with pseudolines **topologically realizable**.
This notion is readily carried over to the corresponding Levi graph of the configuration. Namely, the graph is topologically realizable if it is the Levi graph (=incidence graph) of a configuration of points and pseudolines.

It is known that neither the Fano plane (7_{3}) nor the Moebius-Kantor configuration (8_{3}) are topologically realizable. Among the ten (10_{3}) combinatorial configurations nine are (geometrically) realizable and one is only topologically realizable.

I would like to know what is known about the status of the following complexity decision problem.

Input:Cubic connected bipartite graph G of girth at least 6.

Question:Is G topologically realizable?

The book "Configurations of Points and Lines" by Branko Grunbaum discusses this problem as a classification problem but not as a complexity problem.