  We continue with the investigation began in "The Dixmier conjecture and the
shape of possible counterexamples". In that paper we introduced the notion of
an irreducible pair (P,Q) as the image of the pair (X,Y) of the canonical
generators of W via an endomorphism which is not an automorphism, such that it
cannot be made "smaller", we let B denote the minimum of the greatest common
divisor of the total degrees of P and Q, where (P,Q) runs on the irreducible
pairs and we prove that $\ge $. In the present work we improve this lower bound
by proving that B\ge 15. In order to do this we need to show the the main
results of our previous paper remain valid for a family of algebras
(W^{(l)})_{l\in \mathds{N}} that extend W.
