The minor-approximation conjecture for volume polynomials

Let $k$ be a field. A minor of a volume polynomial is obtained by applying deletions and contractions of variables, and limits are taken in the relevant coefficient space. The minor-approximation conjecture. Every volume polynomial over $k$ is a limit of minors of volume polynomials of convex bodies. This strengthens the field-independence question by proposing that algebraic volume polynomials can be approximated using convex-geometric volume polynomials after passing to minors.