As far as I understood, only the country issuing your license is allowed to put entries in it, correct? So, if you hold a Hungarian license, you have to pass a Hungarian English Proficiency test, correct? Wasn't this the reason why some countries don't offer LP6 exams, because they have no accredited LP6 examiner? There are some exams with mutual agreements between countries, but last time I looked at it, is was almost impossible to say which country does accept which countries exams - you have to check your examiner first.
A remark to the exams. LP 4 and 5 are quite similar (and in many countries a total waste of time for the sake of being able to speak and understand english), with only a few nasty parameters different, while LP6 is something where you are in real conversation. There are even certain places where level 4 and 5 exams can only be thought of the brutal revenge of someone purely obsessed with his image. I found LP6 much nicer, because it is from and for the real world and not synthetic [censored].
Go, try to find a Hungarian level 6 examiner, can't be much of them.