I have no idea if the chlorine will harm it, but I DO know that it will be gone forever if it somehow slips off my hand and that's a risk I'm not willing to take.
I've read countless articles about chlorine destroying white gold. I'm terrible with my jewelry and my rose gold and yellow gold have been exposed to chlorine. Two years worth of damage and both pieces seem fine-- thinking I've already done damage, I went to a jeweler to retip the emerald yellow gold ring and he said it wasn't necessary yet. I would NOT, however, be so nonchalant with white gold.