Alexander's sharp isoperimetric conjecture for minimal submanifolds

Let $\Sigma\subset\mathbb{R}^{n+m}$ be an $n$-dimensional smooth minimal submanifold with smooth boundary $\partial\Sigma$. Write $|\Sigma|$ for its $n$-dimensional volume, $|\partial\Sigma|$ for the $(n-1)$-dimensional volume of its boundary, and let $B^n$ be the open unit ball in $\mathbb{R}^n$. Alexander's conjecture. The sharp inequality

$$|\partial\Sigma|\geq n|B^n|^{\frac{1}{n}}|\Sigma|^{\frac{n-1}{n}}$$

holds, with equality if and only if $\Sigma$ is an $n$-dimensional ball in $\mathbb{R}^n$. This is the codimension-nonzero analogue of the Euclidean isoperimetric inequality; the source attributes it to Alexander, but the supplied text does not state whether it has been resolved.