Seems the idea is supposed to be that it is "backed 1:1 by USD" (option 2) and that instead of changing supply portion of equation (as you mention in scenario one), arbitragers fix it by changing demand (increasing as it goes below $1 and decreasing as it goes above $1). Likely the floating supply changes as well depending on value (e.g: arbitragers buy below $1 reducing supply and hold, then sell above $1, increasing supply).
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