Willmore-energy and continuity-equation conjecture for skew-mean-curvature flow

Let $F_t:\Sigma^n\rightarrow\mathbb R^{n+2}$ be a codimension $2$ submanifold moving by the skew-mean-curvature flow

$$\partial_t q=-JH(q),\qquad q\in\Sigma.$$

Write $\rho=|H|^2$ for the square of its mean curvature. Willmore-energy and continuity-equation conjecture. The following properties are equivalent: (i) the Willmore energy $\mathcal{W}(F_t)$ is invariant; (ii) $\rho$ evolves according to the continuity equation

$$\partial_t\rho+\operatorname{div}(\rho v)=0$$

for some vector field $v$ on $\Sigma$. The conjecture seeks a higher-dimensional analogue of the one-dimensional Hasimoto/Madelung correspondence, relating skew-mean-curvature flow to barotropic fluid equations. Its resolution status is not established by the supplied source.