Seymour's 1-flowing conjecture for matroids

A matroid $M$ is 1-flowing if it is $e$-flowing for every element $e\in E(M)$, where $M$ is $e$-flowing when, for every assignment of non-negative integral capacities to the elements of $E(M)-e$, there is a flow through circuits containing $e$ attaining the minimum capacity of a cocircuit containing $e$. An excluded minor for a minor-closed class is a matroid outside the class whose proper minors all belong to the class. Let $AG(3,2)$ denote the rank-three affine geometry over $\operatorname{GF}(2)$, let $U_{2,4}$ denote the rank-two uniform matroid on four elements, and let $T_{11}$ be the even-cycle matroid obtained from $K_5$ by adding a loop and declaring every edge, including the loop, odd; write $T^*_{11}$ for its dual. Seymour's 1-flowing conjecture. The set of excluded minors for the class of 1-flowing matroids consists of $AG(3,2)$, $U_{2,4}$, $T_{11}$, and $T^*_{11}$. Seymour proved that the class of 1-flowing matroids is minor-closed and that these four matroids are excluded minors. The conjecture asserts that no other excluded minors exist; its resolution is not given in the source.