  Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable
matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected
strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is
bounded relative to $N$, or, up to replacing $M$ by a $\Delta$-$Y$-equivalent
excluded minor, we can choose a pair of elements $\{a,b\}$ such that either
$M\backslash \{a,b\}$ is $N$-fragile, or $M^* \backslash \{a,b\}$ is
$N^*$-fragile.
