no need to run the head stay, it might give you a head gasket problem, motor flexes alot and it pulls on the head, thats when you will get a leak. most guys do not run the head stay.
but bike looks good.
Hmm never heard that before, interesting. Wonder why Suzuki designed it to even have one then? I think I always ran it with it when I used to own it and dont recall having any issues.