Come to think of it, if the dots bled with a little moisture, chances are they will come out with cool water. I think the bigger problem is whether or not the thread used for the cross stitch is colorfast. Many threads that used to be colorfast are not manufactured colorfast anymore because of restrictions on the chemicals that can be used in them.
Theoretically Synthrapol should prevent bleeds from the threads from settling into the fabric. If you do wash in cool water and thread colors bleed, then several washes in hot water with Synthrapol should release the bleeds.