Mulmuley–Sohoni's strengthened permanent versus determinant conjecture

Let $V=\mathbb{C}^{m^2}$, let $\operatorname{det}_m$ be the determinant polynomial on $m\times m$ matrices, and let $\operatorname{perm}_n$ be the permanent of the upper-left $n\times n$ submatrix. Define the orbit closures

$$\mathcal{D}et_m=\overline{\operatorname{GL}(V)\cdot\operatorname{det}_m},\qquad \mathcal{P}erm_n^m=\overline{\operatorname{GL}(V)\cdot x_{m,m}^{m-n}\operatorname{perm}_n}.$$

Mulmuley–Sohoni's strengthened conjecture. For each $c>0$ and infinitely many $n$, $\mathcal{P}erm_n^{n^c}\not\subseteq\mathcal{D}et_{n^c}$. This is a central geometric-complexity-theoretic formulation of the difficulty of representing the permanent by determinants; its resolution remains open.