There is a more specific explanation under Convert:
Special Conditions
If a warrior-class character bearing a weapon-class enhancement is converted, the weapon class enhancement is automatically converted with the character. It converts to the same brigade and remains on the character.
However, I found the following to be quite intriguing, since I do not remember this ruling:
Default Conditions
Special abilities on Heroes (and good weapons)* that have been converted to Evil Characters and evil weapons* can never activate. Special abilities on Evil Characters (and evil weapons)* that have been converted to Heroes may activate, except for special abilities that specifically target Heroes.
Does anyone know if the part about converted good weapons has always been that way?