Can you go swimming with dyed purple hair in a chlorine pool?

Anonymous
Okay I'd ask my hairdresser but it's 10:36 PM and I feel like she might be sleeping.
So as you might of guess my hair is dyed purple.. I have had it dyed for about a month- when I shower with normal water nothing happens but when I shampoo it purple comes out of my hair. Since chlorine is a bleaching agent I'm afraid my hair my fall out or something weird like that..

What will happen if I swim with bleached purple hair in a chlorine pool?

(I am unable able to wear a swimming cap in this senecio)
Can you go swimming with dyed purple hair in a chlorine pool?
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