  Building on recent work of the authors, we use degenerations to chains of
elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank
conjecture, in genus $22$ and $23$. This constitutes a major step forward in
Farkas' program to prove that the moduli spaces of curves of genus $22$ and
$23$ are of general type. Our techniques involve a combination of the
Eisenbud-Harris theory of limit linear series, and the notion of linked linear
series developed by the second author.
