Conjecture that every gaussoid is compatible with all quadrics in J_n

Let $J_n$ be the ideal whose quadrics include the trinomials defining compatibility of gaussoids, and let a gaussoid be a collection of almost-principal minors satisfying the gaussoid axioms. A gaussoid is compatible with a polynomial when the corresponding vanishing and nonvanishing pattern satisfies that polynomial.

All-quadrics compatibility conjecture. Every gaussoid is compatible with all quadrics in $J_n$, not just trinomials.

This conjecture proposes that compatibility with the quadratic relations of $J_n$ is already determined by the trinomial relations. The paper presents it as a direction for developing gaussoid theory in parallel with matroid theory; its resolution is not given here.