I am sure it is incorrect.
There is a grey area (no pun intended) there but it is in a different department: UK night = IFR but the FARs require an FAA PPL to have an IR to fly IFR, so a plain FAA PPL (who has an automatic night privilege, unless done under the concessions like Alaska or Hawaii) cannot, apparently, fly at night over here, or in any other country where night=IFR (which is quite a few of them).
There used to be an easier way to get the CV certificate under FAA, by using their demonstrated ability route involving lights shining from an airport tower. I vaguely recall hearing that the FAA no longer offer this, but could well be wrong. Getting the initial CV certificate (which is valid for life) is indeed a huge and well worn run-around for many wannabe airline pilots who travel to the far corners of the world to get that piece of paper...