Yeah, don't count on drywall for any lateral bracing. You'll have over 2000 lbs of weight on there, so if you and 10 of your large friends got on and shifted a bit......
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Put some diagonal bracing on the stand, on the inside. The bigger the better. On the back and sides would be enough.
I know it may seem like overkill now, but I prefer to think of the stand as it's own unit. I'm thinking of a 2 X 4 in each upper corner, cut at a 45 degree angle on each end, then screwed into the frame.
Mitch