Classical knotoid embedding conjecture in virtual knotoids

A knotoid diagram in $S^2$ is a knotoid diagram on the sphere, and two such diagrams are virtually equivalent when they represent the same virtual knotoid after allowing virtual-knot equivalences. The moves $\Omega$ are the knotoid Reidemeister-type moves used in the source, and isotopy of $S^2$ is ambient isotopy on the sphere.

Classical knotoid embedding conjecture. If two knotoid diagrams in $S^2$ are virtually equivalent, then they are equivalent to each other by finitely many $\Omega$-moves and isotopy of $S^2$.

This would establish that the theory of knotoids in $S^2$ embeds into the theory of virtual knotoids. The source explicitly says that this was unknown at the time and provides no resolution.