The $\mathbb{P}_4$ characterization conjecture for matroid representability

Let $\alpha$ be an indeterminate, and define the partial field

$$\mathbb{P}_4:= \mathbb{P}(\mathbb{Q}(\alpha),\{\alpha,\alpha-1,\alpha+1,\alpha-2\}).$$

$\mathbb{P}_4$ characterization conjecture. A matroid $M$ is representable over all finite fields with at least $4$ elements if and only if $M$ is representable over $\mathbb{P}_4$. The statement is offered as a candidate after the preceding finite-field list conjecture could not be proved even for $k=4$; no resolution is given.