Minimal genus conjecture for minimal crossing diagrams of virtual links

A virtual link is represented by link diagrams considered up to the equivalence allowing stabilization and destabilization of diagrams in thickened surfaces. The crossing number of a diagram is its number of crossings, and the genus is the genus of the supporting surface; a diagram has minimal genus if its supporting surface has the smallest possible genus among diagrams representing the link.

Minimal genus conjecture for virtual links. Given a virtual link $L$, any minimal crossing diagram for it has minimal genus.

The conjecture would rule out minimal-crossing diagrams supported on unnecessarily high-genus surfaces. In the paper it is presented as the missing step that would give a direct proof of the first and second Tait conjectures for virtual links; the supplied text gives no resolution.