The Dixmier conjecture and the shape of possible counterexamples II

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.