The trays do blow the red ink, however there are companies that (illegally) remove the unremoveable marking ink.
Being part of a research group regarding ATM security was a really nice experience for me. It's a different world, full of unexpected things.
Most ATM’s are cheap Hyosung, tranax, etc. they most certainly do not have dye packs.
Source: I used to operate some ATM’s as a side biz.