Tipsy?? A "workload to balance at low speeds (compared to their bikes)?? What the heck does that mean? You need friends that can give you better feedback than "tipsy".
What are their bikes? A ZX6 should present no low speed handling problems, if everything is aligned properly, and in good working order. Has it been crashed? Maybe the forks are tweeked, or the frame is out of alignment? Tire pressure O.K.? Has anyone been monkeying around with the fork height adjustment (raising/lowering forks in the triple clamp)?
Can these guys actually ride? Do they have any clue what they are talking about?