False positive isn't gonna happen, but THC is for some reason more attracted to dark/afro hair, so your hair would probably hold slightly more than a person with light hair with similar smoking habits.

They should only be testing the first 1.5inches of hair anyway, to coveer a 90 day window, there are questions about the accuracy of the test after 90 days so it's not really done under normal circumstances.

Is that laundry detergent you said you bought? I'm in UK so not familiar with brand names. If it is laundry detergent you could try along that with a baking soda paste which is very alkaline and will help to open the cuticle before using a stripping shampoo, Jerry G says that Toxin Wash shampoo is good and can reduce levels without any other treatment.

The best thing really would be to bleach and re-dye along with this, even if you only do it once, I've bleached and redyed 4 times now in the last 3 weeks, depends how important it is for you to pass I guess