The 3-core conjecture for maximal typical rank

Let $G$ be a graph with a non-empty $3$-core, meaning its $3$-core as defined in the paper is non-empty. Let the maximal typical rank of $G$ be the largest rank occurring typically among its matrix completions.

3-core conjecture. The maximal typical rank of $G$ is at least $3$.

This conjecture would imply the characterization problem for planar bipartite graphs having $3$ as a typical rank, and more generally for bipartite graphs with generic completion rank $2$. The source does not report a resolution.