Characterization of finite fields representing all graphic delta-matroids

Let $\Gamma$ be a finite abelian group of order at least $2$, and let $\mathbb{F}$ be a finite field. A $\Gamma$-graphic delta-matroid is a delta-matroid associated with a $\Gamma$-gain graph; it is representable over $\mathbb{F}$ when it admits a representation over $\mathbb{F}$. Complete characterization conjecture. Every $\Gamma$-graphic delta-matroid is representable over $\mathbb{F}$ if and only if

$$(\Gamma,\mathbb{F})=(\mathbb{Z}_p^k,\operatorname{GF}(p^\ell))$$

for some prime $p$ and positive integers $k\leq\ell$. The preceding results establish the sufficiency when $k\leq\ell$ and show that necessity requires $\Gamma$ to be an elementary abelian $p$-group and $\mathbb{F}$ to have characteristic $p$; the conjecture asserts that these conditions, together with $k\leq\ell$, give the complete characterization.