Mulmuley–Sohoni's orbit-closure conjecture for permanent versus determinant

Let $\ell$ be a linear coordinate on $\mathbb C^1$, and define the projective orbit closures

$$\mathcal{Det}_n:=\overline{GL_{n^2}\cdot[\operatorname{det}_n]},$$ $$\mathcal{Perm}^m_n:=\overline{GL_{n^2}\cdot[\ell^{n-m}\operatorname{perm}_m]}.$$

Let $n(m)$ be a polynomial. Mulmuley–Sohoni's orbit-closure conjecture. For all sufficiently large $m$,

$$\mathcal{Perm}^m_{n(m)}\not\subset\mathcal{Det}_{n(m)}.$$

This stronger geometric formulation asks for separation of orbit closures rather than only separation from the determinant orbit. It remains open and is central to geometric complexity theory.