The set-theoretic version of the Salmon Conjecture (that is, finding the equations that cut out the fourth secant variety of the Segre embedding of $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ set-theoretically) has been resolved by S. Friedland and E. Gross in https://arxiv.org/abs/1104.1776

In view of this, I am wondering if resolving the scheme-theoretic version of the Salmon conjecture (that is, finding the defining ideal of the above variety) 1) is much harder/needing new ideas compared to the set-theoretic solution and 2) carries more information for the biological applications that motivated the conjecture than the set-theoretic version.

Also, is the problem of finding the ideals of other secant varieties of Segre embeddings 1) expected to be resolved in the near future and 2) relevant for biological or other applications? What are some approachable/interesting problems here?