The K-contact energy-minimization conjecture for curl eigenforms

Let $\eta$ be the curl eigenform defined by a $K$-contact structure, and let $\Xi_\eta$ denote the associated coadjoint orbit. An eigenform is energy-minimizing on $\Xi_\eta$ if it minimizes the relevant energy functional among forms in that orbit.

K-contact energy-minimization conjecture. The curl eigenform $\eta$ defined by a $K$-contact structure is always energy-minimizing on $\Xi_\eta$.

The conjecture arises because the form $\eta$ is identified as a natural candidate for the principal curl eigenform, while the supplied context notes that it need not always be principal. The source gives no resolution of the universal energy-minimization assertion.