Boundary length decrease conjecture for higher-dimensional free boundary minimal submanifolds

Let $B^n$ be the unit ball with boundary $S^{n-1}$. Let $\Sigma$ be a $k$-dimensional minimal submanifold in $B^n$ with boundary $\partial\Sigma\subset S^{n-1}$, and suppose that the conormal vector of $\Sigma$ equals the position vector. For a conformal transformation $f$ of the ball, write $|f(\partial\Sigma)|$ for the boundary volume of the image. Higher-dimensional boundary length decrease conjecture. For every conformal transformation $f$ of $B^n$,

$$|f(\partial\Sigma)|\leq |\partial\Sigma|.$$

The claim extends the proved two-dimensional boundary length inequality to higher-dimensional free boundary minimal submanifolds; the source notes evidence for cones and a second-order boundary-volume decrease, but does not state a resolution.