  Let $f$ and $g$ be permutable transcendental entire functions. We use a
recent analysis of the dynamical behaviour 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 $J(f)=J(g)$ provided that
neither $f$ nor $g$ has a simply connected wandering domain in the fast
escaping set.
