The conjecture that planar-move graphs differ from knot-move graphs

Let ${\cal C}_1$ be the set of graphs isotopic to planar graphs, ${\cal C}_2$ the set of graphs isotopic to graphs obtained from planar graphs by the specified moves, ${\cal C}_3$ the set of graphs isotopic to graphs obtained from knots, including the unknot, by those moves, and ${\cal C}_4$ the set of graphs whose polynomial $P$ equals $1$. The known inclusions are ${\cal C}_1\subseteq{\cal C}_2\subseteq{\cal C}_3\subseteq{\cal C}_4$.

The conjecture that ${\cal C}_2$ and ${\cal C}_3$ differ.

$${\cal C}_2\neq{\cal C}_3.$$

The question concerns whether the classes generated from planar graphs and from knots by the allowed graph moves coincide; the source gives the preceding inclusions and an example proving ${\cal C}_1\neq{\cal C}_2$, but no resolution of this separation is supplied.