Excluded-minor conjecture for 3-regular matroids

Let $M$ be a matroid. A matroid is $3$-regular if it has a representation over $\mathbb{Q}(\alpha_1,\alpha_2,\alpha_3)$ in which every non-zero subdeterminant is an integer power of a difference of distinct members of $\{0,1,\alpha_1,\alpha_2,\alpha_3\}$. Let $\Delta^{(*)}(U_{2,7})$ denote the family of six matroids obtained from $U_{2,7}$ by $\Delta$-$Y$ exchange and dualising. Excluded-minor conjecture for $3$-regular matroids. A matroid $M$ is $3$-regular if and only if it has no minor isomorphic to one of the following $33$ matroids:

$$F_7, F_7^-, F_7^=, H_7, M(K_4)+e, \mathcal{W}^3+e, \Lambda_3, Q_6+e, P_6+e, U_{3,7},$$

and their duals; a matroid in $\Delta^{(*)}(U_{2,7})$; and $\mathit{AG}(2,3)\backslash e$, $(\mathit{AG}(2,3)\backslash e)^*$, $(\mathit{AG}(2,3)\backslash e)^{\Delta Y}$, $P_8$, $P_8^-$, $P_8^=$, and $\mathit{TQ}_8$. The conjecture would provide the excluded-minor characterisation for $3$-regular matroids, complementing the proved characterisation for $2$-regular matroids; the listed excluded minors are known up to size $13$, while completeness of the list remains open.