According to the website W.T. Tutte has also proven this, but I wasn't able to find any reference to his proof. We can however assume that it is true, because of the proof in [Zhang].
F. Jaeger used the above result in combination with is 4-flow theorem to proof the conjecture for 4-edge-connected graphs.
3-edge colorable graphs
Since a cubic graph admits a nowhere-zero 4-flow if and only if it is 3-edge-colorable, the conjecture holds for 3-edge-colorable cubic graphs.
By using vertex splitting arguments, it can be shown that we can reduce the problem to the set of cubic graphs that are not 3-edge colorable.
Bermond, Jackson and Jaeger used Jaeger's 8-flow theorem to prove that graphs with no cut-edge have a list of circuits so that every edge is contained in exactly four.
Using Seymour's 6-flow theorem, Fan proved that graphs with no cut-edge have a list of circuits so that every edge is contained in exactly six.