GCT multiplicity-obstruction conjecture of Mulmuley and Sohoni

Let $\mathrm{dc}(\mathrm{per}_m)$ be the determinantal complexity of the $m\times m$ permanent. For integers $d,n,m$, let $\gamma_{\lambda,d,n,m}$ and $\delta_{\lambda,d,n}$ denote the multiplicities of the irreducible representation indexed by $\lambda$ in the coordinate rings of the padded-permanent and determinant orbit closures, respectively. GCT multiplicity-obstruction conjecture. There exist multiplicity obstructions showing that

$$\mathrm{dc}(\mathrm{per}_m)>m^c$$

for every constant $c$; namely, for every $n=O(m^c)$ there exists an integer $d$ and a partition $\lambda\vdash dn$ such that

$$\gamma_{\lambda,d,n,m}>\delta_{\lambda,d,n}.$$

Such an obstruction would rule out containment of the padded-permanent orbit closure in the determinant orbit closure and yield lower bounds for the permanent. The source gives no resolution.