Arnold–Khesin coadjoint-orbit density conjecture

Let $X$ be a vector field in the setting of three-dimensional volume-preserving flows, and let its corresponding helicity level set be the set of vector fields with the same helicity as $X$. A subset is somewhere dense if its closure has nonempty interior in some topology. Arnold–Khesin's coadjoint-orbit conjecture. There exists a vector field $X$ whose coadjoint orbit is somewhere dense in its corresponding helicity level set, for some topology. The conjecture concerns whether a coadjoint orbit can be large within a helicity level set. The source attributes it to Arnold and Khesin, but supplies no resolution status or precise topology.