Surface bracket state characterization of virtual closure images

Let $K$ be a virtual knot of genus one, and let a torus representation be a representation of $K$ by a knot diagram in the thickened torus. The surface bracket states of such a representation are the state curves obtained from its surface bracket expansion; their multiplicity is the coefficient with which the corresponding homology class occurs. Let $\overline{v}$ denote the virtual closure map from knotoids to virtual knots. Surface bracket state characterization conjecture. A virtual knot of genus one lies in the image of the virtual closure map if and only if all of the surface bracket states of any of its torus representations have multiplicity at most one. The conjecture proposes a direct surface-bracket criterion for recognizing virtual knots in the image of the virtual closure map, equivalently the genus-one knots of geometric degree one; the source provides no evidence that it has been resolved.