There's been a couple of successfully restored periwinkle bags discussed recently and both had colour loss. If you do a search in this thread you will find all helpful the details.

Welcome and Good luck!

Your bag will be gorgeous!
Welcome to vintage!
@whateve taught us three of the most important pieces of information about white bags and those are:
1) it seems white bags are dyed differently to coloured bags in that they are rather more painted than dyed. (I may not be describing this perfectly correctly, so other opinions are more than welcome).
2) some stains/marks on white bags can be remedied by spot treatments with Acne Cream containing benzoyl peroxide. Put a dab of cream on the mark and leave for several hours. You may need to repeat a couple of times
3) if the mark is gray or doesn't change with Acne Cream it could be the bare leather showing through. In that case painting/dyeing is required
Use the search feature in this thread for a treasure trove of information on restoring/painting white bags.

Good luck and keep us posted!