Conjecture that the quadrics generate the gaussoid ideal J_n

Let $J_n$ be the prime ideal associated with the Gaussian conditional-independence model, and suppose its quadrics are the relations listed explicitly in Theorem~.

Quadratic generation conjecture. The ideal $J_n$ is generated by its quadrics, listed explicitly in Theorem~.

The claim concerns the defining equations of the Gaussian model and would give a complete generation statement from the explicitly described quadratic relations. The source does not state that this conjecture has been resolved.