Hessian discriminant

Anna Seigal [see 1] identifies the Hessian discriminant as a locus where the complex rank of $f$ jumps from $5$ to $6$. This is a hypersurface of degree $120$ in $\mathbb{P}^{19}$, invariant under $\textrm{PGL}(3)$. How to write the Hessian discriminant in terms of fundamental invariants?

This problem is currently being worked on by Rodica Dinu and Tim Seynnaeve. We believe that $f = I_{40}^3$.

1. Seigal, A. Ranks and Symmetric Ranks of Cubic Surfaces.2018.//[https://arxiv.org/abs/1801.05377]
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