No it's not true is the short answer......but the longer answer is it all depends on which country you hunt in, for example Mozambique didn't apply to CITES to downlist their Elephants when they should have done. Hence, the current situation is that hunters cannot get import licences to import any Elephant products. This rule seems to me to be most strictly adhered to when importing into the USA.

Another factor is whether you hunt with a reputable outfitter who has the correct licences for you to take the animals in the first place. For Example, if you book a hunt Tanzania or Botswana for an Elephant or a Leopard, your CITES people should pre-issue you with the relevant CITES import tag, and you go from there.

I am of course assuming that you're not planning to hunt anything on the wrong list. You'll find full details on the CITES website.