hello @NonoyÀrmenia .. have you tried checking IMEI #? compare the IMEI # on the box, underneath the battery as well as the IMEI # on the actual phone located at "about device..
then, try the LCD test..
Have you tried the *#1234# method? If it does work, compare with the about device and check every command like *#2263# if it's LTE supported. Usually LTE can't be faked, except if that's an i9500 which octacore that doesn't have LTE. But if it's a i9505 continue. Also try other commands like *#0011# *#06# *#0*# and so-on.
If none of those codes work particularly the specific Samsung commands. It's fake.
Commands are issue via Keypad (where you input numbers to call).
And did you ask your wife if she received the phone with the label unbroken?