score:5
And your question? your source clearly stated it was created, as a radio telegraphy service, in 1938.
An undersea cable wasn't laid, as the same source states. And by the time telegraph cables were already being replaced by wireless telegraphy services (or radio telegraphy as the Americans called it).
Wikipedia states that the first cable from the US across the Pacific to a point further away than Hawaii was only taken into service in 1991, and that one ran to Japan. See [here].1
This is a nice map of undersea cables. I don't know how complete it is, but it shows indeed no direct cable link between the US mainland and Australia (there appears to be one from Hawaii to Australia).
So no, there is no direct telegraph cable between the US mainland (or indeed anywhere in the Americas) and Austrial and never was.