The intrinsic virtual knotting conjecture for intrinsically knotted graphs

Let $G$ be a graph. Call $G$ intrinsically knotted if every classical spatial embedding of $G$ contains a non-trivial knot, and call it intrinsically virtually knotted of degree 1 if every virtual diagram of $G$ with at most one virtual crossing contains a virtually knotted cycle. Intrinsic virtual knotting conjecture. If $G$ is intrinsically knotted, then $G$ is intrinsically virtually knotted of degree $1$.

The conjecture is motivated by the claim that the Jones polynomial distinguishes the unknot. The stated theorem proves the conclusion under the additional hypothesis that every diagram of $G$ contains a knot with non-trivial Jones polynomial; the unrestricted assertion remains open in the supplied text.