Permutable entire functions and multiply connected wandering domains

Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behavior in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g   are equal; in particular, we show that J(f)=J(g)J(f)=J(g) provided that neither f nor g has a simply connected wandering domain in the fast escaping set.