The hyperdeterminantal generation conjecture for principal minors of symmetric matrices

Let $Z_n$ be the variety of principal minors of symmetric $n\times n$ matrices, and let $HD$ denote the hyperdeterminantal module, namely the span of the $\left(\operatorname{SL}(2)^{\times n}\right)\ltimes\mathfrak{S}_n$-orbit of the $2\times2\times2$ hyperdeterminant. Hyperdeterminantal generation conjecture. The prime ideal of $Z_n$ is generated in degree four by $HD$ for all $n\geq3$. This conjecture seeks a uniform description of the defining equations of the principal-minor variety; the degree-four generators are known in the case $n=4$, while the assertion for all $n\geq3$ remains open here.