This is the matter I had proved, and same proof.
There is a proceeding of AMS about "nearly abelian polynomials",
http://www.ams.org/journals/proc/2005-133-01/S0002-9939-04-07669-5/S0002-9939-04-07669-5.pdf
And I already published a proceeding of Japan Math. Soc,
https://projecteuclid.org/download/pdf_1/euclid.pja/1195510075
These are same proofs.