Projective-equivalence conjecture for the sixth Terracini locus of a triple Segre variety

Let $X$ be the Segre embedding of $\mathbb{P}^3\times\mathbb{P}^3\times\mathbb{P}^3$ in $\mathbb{P}^{63}$, and let $\pi_i:\mathbb{P}^3\times\mathbb{P}^3\times\mathbb{P}^3\to\mathbb{P}^3$ be the three natural projections. For a subset $S\subset X$ of cardinality $6$, write $\pi_i(S)$ for its image under the $i$-th projection.

Projective-equivalence conjecture. The subsets $S\in X$ of cardinality $6$ such that $\pi_i(S)$ and $\pi_j(S)$ are projectively equivalent, for some fixed $i\neq j$, fill a dense open set in the $6$-th Terracini locus of $X$.

This conjecture describes a dense open part of the Terracini locus for the Segre variety $\mathbb{P}^3\times\mathbb{P}^3\times\mathbb{P}^3$. The surrounding example motivates the claim via families of elliptic normal curves and non-unique decompositions, but the supplied text does not state whether the conjecture has been proved or disproved.