  We consider the random conductance model, where the underlying graph is an
infinite supercritical Galton--Watson tree, the conductances are independent
but their distribution may depend on the degree of the incident vertices. We
prove that, if the mean conductance is finite, there is a deterministic,
strictly positive speed $v$ such that $\lim_{n\to\infty} \frac{|X_n|}{n}= v$
a.s.\ (here, $|\cdot|$ stands for the distance from the root). We give a
formula for $v$ in terms of the laws of certain effective conductances and show
that, if the conductances share the same expected value, the speed is not
larger than the speed of simple random walk on Galton--Watson trees. The proof
relies on finding a reversible measure for the environment observed by the
particle.
