Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functions f:Cⁿ →Cⁿ with det f(x)=1 at all x εCⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open. Without loss of generality assume f(0)=0 and f'(0)=I. We study the existence of certain mappings h_λ, λ > 1, defined by power series in a ball with center at the origin, such that h′_λ(0)=I and h_λ(λf(x))=λh_λ(x). So each h_λconjugates λf to its linear part λI in a ball where it is injective. We conjecture that for Keller's functions f of the homogeneous form f(x)=x +g(x), g(sx)=s^dg(x), g′(x)ⁿ=0, xεCⁿ, sεC the conjugation h_λ for λf is an entire function.