Not necessarily Arch. It would not fade to black because black can have a hue(color) obscured by its value(darkness). If a freshly dyed garment is light enough to be a deep hue(that is, not black/gray/white), then it does not contain enough of the other pigments to be black. It would fade to gray,assuming the garments natural color is white. Being that the other pigments would also fade somewhat, it would end up being of an even lighter value than what the dye it started as.
In short, if a dye has enough of the other pigments to be black, it will be incapable of showing a strong hue because there is not a single pigment with enough presence, even in the absence of all others. If a dye is a strong hue, it will be incapable of becoming black for the same reason. Chemical changes aside.
Example: a dyed garment's dye has all pigments equally. It is black. Remove all pigments but red. The dye is red, but the garment is pink because it no longer has enough dye to cover the natural garment color (white). If the dye is reapplied, the garment will be a deep red.
Example2: A dyed garment is dark red. Its dye is mostly red with traces of all other pigments(IE a natural dye) Remove red. The dye is near black with a slight green hue. The garment is light gray with a slight greenish hue.
The fact that the dye loses a color after it is applied to the garment changes a lot.