The second fundamental theorem conjecture for the determinant orbit closure

Let $v$ be the determinant function and let $\Delta_V[v]$ be the class variety associated with $NC$, namely the orbit closure of the determinant. Let $X(Pi_v)$ be the variety and scheme defined by the ideal generated by the representation-theoretic modules in $Pi_v$. Second fundamental theorem conjecture. One has

$$X(Pi_v)=\Delta_V[v]$$

as a variety, and more strongly, as a scheme. This asks whether the ideal of the determinant orbit closure is generated by these representation-theoretic equations; the analogous assertion is known for homogeneous varieties such as $G/P$, but the determinant orbit closure is a central unresolved case in geometric complexity theory.