Here is a counterexample to the original question:

In retrospect, this should have been found earlier. It remains to find a 3-connected counterexample and to establish the complexity of finding the chromatic index of these graphs

Here is an example that shows the answer to original question is no. The graph shown has chromatic index 5.

In retrospect, this should have been found earlier. It remains to find a 3-connected counterexample and to establish the complexity of finding the chromatic index of these graphs