Wulff inequality for minimal submanifolds

Let $\mathcal{W}$ be the Wulff shape associated with a positive anisotropic function $\Phi$, and let $\Sigma$ be an $n$-dimensional minimal submanifold in $\mathbb{R}^{n+m}$. For each $n$-dimensional affine subspace $P$, let $\operatorname{proj}_P\mathcal{W}$ denote the projection of $\mathcal{W}$ to $P$. Choose an $n$-dimensional affine subspace $\overline{P}$ minimizing the projected volume, and set

$$W^*=\operatorname{proj}_{\overline{P}}\mathcal{W},\qquad |W^*|=\min\{ |\operatorname{proj}_P\mathcal{W}|:P\in Gr_n(\mathbb{R}^{n+m})\}.$$

Wulff inequality for minimal submanifolds. The inequality

$$\frac{P_{\Phi}(\Sigma)}{|\Sigma|^{\frac{n-1}{n}}}\geq\frac{P_{\Phi}(W^*)}{|W^*|^{\frac{n-1}{n}}}$$

holds, with equality if and only if $\Sigma$ is homothetic to some $W^*$. This is the proposed nonzero-codimension extension of the classical Wulff inequality. The supplied text presents it as a natural question and gives no resolution, while noting that related anisotropic minimal-submanifold inequalities have been studied.