  Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable
matroids for some partial field $\mathbb{P}$, let $N$ be a $3$-connected strong
$\mathbb{P}$-stabilizer that is non-binary, and suppose $M$ has a pair of
elements $\{a,b\}$ such that $M\backslash a,b$ is $3$-connected with an
$N$-minor. Suppose also that $|E(M)| \geq |E(N)|+11$ and $M \backslash a,b$ is
not $N$-fragile. In the prequel to this paper, we proved that $M \backslash
a,b$ is at most five elements away from an $N$-fragile minor. An element $e$ in
a matroid $M'$ is $N$-essential if neither $M'/e$ nor $M' \backslash e$ has an
$N$-minor. In this paper, we prove that, under mild assumptions, $M \backslash
a,b$ is one element away from a minor having at least $r(M)-2$ elements that
are $N$-essential.

Combinatorics