Mulmuley–Sohoni padded-permanent border determinantal complexity conjecture

For $n>m$, let $X_{11}^{n-m}\mathrm{per}_m$ be the padded permanent, viewed as an element of $\mathsf{Sym}^n\mathbb{C}^{n^2}$, and let ${{\mathcal{D}}et}_n$ be the Euclidean (equivalently, Zariski) closure of the $\mathrm{GL}_{n^2}$-orbit of $\mathrm{det}_n$. Mulmuley–Sohoni's border determinantal complexity conjecture. For every $c\in\mathbb{N}_{\ge 1}$, one has

$$X_{11}^{m^c-m}\mathrm{per}_m\notin{{\mathcal{D}}et}_{m^c}$$

for infinitely many $m$. Membership of the padded permanent in this orbit closure would imply an upper bound on its determinantal complexity, so this conjecture would imply Valiant's determinantal complexity conjecture. It was proposed as a geometric-complexity-theory approach and remains open in the stated source context.