  Arrondo, Sols and De Cataldo proved that there are only finitely many
families of codimension two subvarieties not of general type in the smooth
quadric of dimension $n+2$ for $n\ge 2 $, $n\neq 4$. In this paper we drop the
assumption $n\neq 4$ from the previous result (obviously the assumption $n\ge
2$ cannot be removed).
