I did a literature search on chlorhexidine (active ingredient in your mouthwash) catalyzed degradation of Nylon and found nothing.
Looking at the structure of the molecule, I'd say it is unlikely to damage Nylon.
However, your subject line says bleach. Bleach, especially if highly concentrated, is bad for Nylon. If your rig has been stewing in bleach in your trunk for who knows how long, that's very bad.
If you are seeing evidence of corrosion of the metal grommets (due to the white precipitate being hard to remove,) proceed with serious caution.
Get your final answer from your rigger. He knows more about this equipment than I do. I'm just adding my two cents.