  Let $C$ be a curve and $L$ a very ample line bundle. The Green-Lazarsfeld
Secant conjecture predicts that if the degree of $L$ is at least
$2g+p+1-2h^1(C,L)-Cliff(C)$ and if, in addition, $L$ is $p+1$ very ample, then
the Koszul group $K_{p,2}(C,L)$ vanishes. In this article, we establish the
conjecture in the extremal case, i.e.\ the case where the degree is exactly
$2g+p+1-2h^1(C,L)-Cliff(C)$, subject to explicit genericity assumptions on $C$
and $L$. In particular, the gonality of $C$ is allowed to be arbitrary (in our
cases $gon(C)=Cliff(C)+2$).
