The fundamental-elements conjecture for universal partial fields

Let $N$ be a matroid and let ${\mathbb{P}}_N$ denote its universal partial field. A ${\mathbb{P}}_N$-representation of $N$ is a representation over ${\mathbb{P}}_N$, and two such representations are equivalent in the usual sense. Fundamental-elements conjecture. If ${\mathbb{P}}_N$ has finitely many fundamental elements, then all ${\mathbb{P}}_N$-representations of $N$ are equivalent.

The conjecture links the finiteness of the fundamental elements of a universal partial field with uniqueness of the corresponding matroid representations. Its status is not resolved in the source.