I made a little google search and found this paper, that mentions that the conjecture holds true for polynomials and special functions Cohen calls "full functions". I'm not able to download the paper though. Cohen's paper is: H. Cohen, On fixed points of commuting functions, Proc. Amer. Math. Soc. 15. (1964),
An idea that occured to me, since apparently this is true for polynomials. Can we not use some form of approximation theorem (Stone-Weierstrass or whatever there is) to conclude the result for other continuous functions between the closed unit balls?