The field-independence conjecture for volume polynomials

Let $k$ be a field, and let $\mathbb{V}^d_n(\mathbb{R},k)$ denote the set of volume polynomials over $k$. The field-independence conjecture. The set of volume polynomials over $k$ is independent of the choice of $k$. The paper notes that independence is known in some low-dimensional or characteristic-dependent senses, but is not known in general when $d\geq3$ and $n\geq3$.