Degree-12 equations for the variety of principal minors

Let $\mathcal{V}_n\subset\mathbb{P}^{2^n-1}$ be the projective variety parametrizing principal minors of an $n\times n$ matrix.

Degree-12 equations conjecture. The variety $\mathcal{V}_n$ is cut out by equations of degree $12$.

This appears immediately after the paper discusses the possibility that a set-theoretic description of the principal-minor variety may be within reach. The supplied text does not specify whether this means that all defining equations can be chosen in degree $12$ or that degree-$12$ equations suffice set-theoretically, so the precise intended formulation should be checked against the surrounding source.