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Would ultrasonic theoretically work?
Ultrasonic is a good way to scrub rigid parts without using aggressive chemistry. I haven't really looked at how well it works on textiles.
This identifies the problems which sprung to my mind, it looks like there are solutions but whether the domestic products actually have them is not answered.
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Dilute some bleach (couple of capfuls) in the sink, soak socks in this for 20mins then wash in machine. I've also done this by pouring bleach into the detergent tray of the washing machine, but I did ruin some of mrs C's clothes using the direct injection to washing machine method so not 100% recommended.
FWIW, my Mavic white socks are much better at staying white than any of the others.
The problem with any chemistry which is good at making your laundry white is that it also tends to kill your garments quicker, so there's a trade off.