Uniform partial-field characterization conjecture for matroids over extensions of $\operatorname{GF}(2)$

Let $\mathbb{U}_1^{(2)}$ be the partial field

$$\mathbb{U}_1^{(2)}:= \mathbb{P}(\operatorname{GF}(2)(\alpha),\{\alpha,1+\alpha\}).$$

Uniform partial-field characterization conjecture. A matroid is representable over $\operatorname{GF}(2^k)$ for all $k>1$ if and only if it is representable over $\mathbb{U}_1^{(2)}$. The conjecture is described as lying just outside the scope of the Lift Theorem, and no resolution is given.