Grad's conjecture for toroidally nested MHD equilibria

Let $M \subset \mathbb{R}^3$ be a toroidal domain. An MHD equilibrium on $M$ is a pair $(X,p)$ satisfying the stationary ideal magnetohydrodynamic equilibrium equations, where $p$ is the pressure function. The pressure has toroidally nested level sets when its regular level sets are nested tori in $M$. The equilibrium is axisymmetric when there is an infinitesimal generator of rotations about an axis preserving $M$ whose induced vector field is a Killing symmetry of both $X$ and $p$.

Grad's conjecture. If there exists an MHD equilibrium $(X,p)$ on $M$ whose pressure has toroidally nested level sets, then the equilibrium is axisymmetric.

This is the classical conjecture that toroidally nested MHD equilibria in a toroidal Euclidean domain must possess Euclidean rotational symmetry. The paper contrasts this conjecture with its generic asymmetry results for adapted metrics; whether the Euclidean assertion holds remains the question under discussion.