Number of homogeneous components of counterexamples to the Dixmier conjecture

Assume that P and Q are elements of A₁ satisfying $[P,Q] = 1$. The Dixmier Conjecture for A₁ says that they always generate A₁. We show that if P is a sum of not more than~$4$ homogeneous elements of A₁ then P and Q generate A₁, which generalizes the main result in arXiv:2210.00257.