Hyperdeterminantal module generation conjecture for the symmetric principal-minor ideal

Let $P_n$ be the prime ideal of the algebraic variety that is the image of the principal-minor map for complex symmetric $n\times n$-matrices. Let the hyperdeterminantal module be the orbit of the $2\times2\times2$ hyperdeterminant under the action of the relevant group $G$ and the symmetric group $S_n$. The hyperdeterminantal module generation conjecture says that $P_n$ is generated by the hyperdeterminantal module. This is a representation-theoretic strengthening of the converse problem for hyperdeterminantal relations: the paper presents it as a natural conjecture after resolving the stated symmetric principal-minor assignment problem, but does not provide a resolution of this ideal-generation claim.