The level partial-field universality conjecture

Let ${\mathbb{P}}$ be a partial field, and let ${\cal M}$ be a class of matroids. Say that ${\mathbb{P}}$ is ${\cal M}$-universal if, for every partial field ${\mathbb{P}}'$ such that every matroid in ${\cal M}$ is ${\mathbb{P}}'$-representable, there exists a homomorphism ${\mathbb{P}}\rightarrow {\mathbb{P}}'$. A partial field is level if it is obtained as the universal partial field associated with the class of matroids representable over another partial field. Level partial-field universality conjecture. Let ${\cal M}$ be the set of all ${\mathbb{P}}$-representable matroids, where ${\mathbb{P}}$ is a level partial field. Then ${\mathbb{P}}$ is ${\cal M}$-universal.

This conjecture asks whether every level partial field is universal for precisely the class of matroids it represents. The source presents it as an open question in the discussion of universal and level partial fields.