I have a good experience of recently using the CX customer service number (at the bottom of the website) for the United States. For Japan Airlines FFP, I would only be able to earn mileage / qualifying points for booking class L and above.

I called the number around 6 PM EST yesterday and the automated number originally advised me that I would have a 10 minute waiting period, but I was speaking to an agent within 2 minute's time. After specifying the date, route numbers, and specifying that I want to have the fare class L, the agent confirmed my name, date of birth, flight details, and also sent an email to confirm the informations.

Once I confirmed the information was correct, I then got a link via email again to purchase my fare. Sure enough, the fare class L was there on the e-ticket PDF. I would recommend their customer service if needed and the phone call was no more than 15 minutes.

Altlough, I wish CX would still peg their fares to specific fare class and associated benefits. On Japan Airlines website, you have the choice of the cheapest (of what's available) fare classes, upgradable, and then fully flexible options, all which have different accumulation points based on fare class for your oneworld programs.