Planar graph detection conjecture via ribbon graph homology

Let $(G,S)$ be a ribbon graph embedded in $S^3$, and let $H_i$ denote its homology groups. Planar graph detection conjecture. The ribbon graph $(G,S)$ is isotopic to a planar graph if and only if

$$H_i=0\quad\text{for }i<0,$$

and $H_0$ is free of rank $2$. This proposes a homological criterion for detecting planar embeddings among ribbon graphs; the source gives no resolution, so the conjecture remains open.