Multilinear representation conjecture for skew partial fields

A skew partial field is a partial field whose multiplicative group need not be commutative. Let $\mathbb{P}(n,\mathbb{F})$ denote the partial field associated with $n\times n$ matrices over a field $\mathbb{F}$, and let a partial-field homomorphism preserve the partial-field structure. Multilinear representation conjecture. For every skew partial field $\mathbb{P}$ there exists a partial-field homomorphism

$$\mathbb{P}\rightarrow \mathbb{P}(n,\mathbb{F})$$

for some integer $n$ and field $\mathbb{F}$. This would characterize matroids representable over skew partial fields as precisely those having multilinear representations over fields; the paper reports this as an open problem motivated by its examples.