The contamination comes when the sponge in the capping station touches the head, and creates a path for the inks to actually pull from one cart. into the other cart.), most of the time only as far as the damper.
A few powerful cleanings will do it, or using a syringe, as you suggested.
Then, check your capping station sponges to make sure it doesn't happen again. They need to be below the rubber seal surrounding them, and not really wet.
Sometimes, a person will fill the sponge with cleaning fluid to soak the head overnight. that can also cause the sponge to swell and create the right conditions for contamination.