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You don't specify whether or not you will have risers in addition to the treads. If so, they will add considerable stiffness to the stairs. Rather than make the center support out of plate, I would suggest you make it in a box section. The wider and deeper, the better. If it contains internal bulkheads, so much the better. I would use a structural adhesive on any mechanically fastened connection. You can also use the railing on both sides to substantially stiffen the structure, depending on how it's constructed.
I don't know if you are building this in a permitted project, you might consider getting an engineers stamp on whatever design you come up with.
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